Cyclic behaviour of short glass fibre reinforced polyamide: Experimental study and constitutive equations

International Journal of Plasticity 27 (2011) 1267–1293

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International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Cyclic behaviour of short glass fibre reinforced polyamide: Experimental study and constitutive equations A. Launay a,b,c,⇑, M.H. Maitournam a, Y. Marco b, I. Raoult c, F. Szmytka c a

Laboratoire de Mécanique des Solides (CNRS UMR 7649), École polytechnique, 91128 Palaiseau, France Laboratoire Brestois de Mécanique et des Systèmes (EA 4325 ENSIETA/UBO/ENIB), 2 rue François Verny, 29806 Brest Cedex 9, France c PSA Peugeot Citroën (Direction Scientifique et des Technologies Futures), Route de Gisy, 78943 Vélizy-Villacoublay Cedex, France b

a r t i c l e

i n f o

Article history: Received 26 November 2010 Received in final revised form 19 January 2011 Available online 18 February 2011 Keywords: Cyclic loading Constitutive behaviour Elasto-viscoplastic material Polymeric material Mechanical testing

a b s t r a c t Polymer matrix composites are widely used in the automotive industry and undergo fatigue loadings. The investigation of the nonlinear cyclic behaviour of such materials is a required preliminary work for a confident fatigue design, but has not involved many publications in the literature. This paper presents an extensive experimental study conducted on a polyamide 66 reinforced with 35 wt% of short glass fibres (PA66 GF35), at room temperature. The material was tested in two conditions: dry-as-moulded (DAM) and at the equilibrium with air containing 50% of relative humidity (RH50). An exhaustive experimental campaign in tensile mode has been carried out, including various strain or stress rates, complex mechanical histories and local thermo-mechanical recordings. Such an extended database allowed us to highlight several complex physical phenomena: viscoelastic effects at different time scales, irrecoverable mechanisms, nonlinear kinematic hardening, non-linear viscous flow rule, cyclic softening. Taking into account this advanced analysis, a constitutive model describing the cyclic behaviour is proposed. As the experimental database only includes uniaxial tensile tests, the general 3D anisotropic frame is reduced to an uniaxial model valid for a specific orientation distribution. The robust identification process is based on tests which enable the uncoupling between the underlined mechanical features. This strategy leads to a model which accurately predicts the cyclic behaviour of conditioned as well as dry materials under complex tensile loadings. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction In a context of CO2 emission reduction, the automotive industry increasingly uses plastic materials to take advantage of their light weight and their complex mould designs. Polymer matrix composites (PMCs) and especially short glass fibre reinforced (SGFR) thermoplastics exhibit the required stiffness for structural applications (such as intake manifolds, inlet gas compressor exit, engine mount limiter) thanks to the glass fibres. The choice of a polyamide matrix provides a good thermal strength for a moderate cost. Those components undergo cyclic loadings during their service life, induced by mechanical (pulsed pressure) as well as environmental sources (temperature, humidity). One of the main issues for the engineers therefore lies in the prediction of the fatigue life duration under complex loadings. Two steps are classically involved in solving the problem. On the one hand,

⇑ Corresponding author at: PSA Peugeot Citroën – Direction de la Recherche et de l’Ingénierie Avancée, VV1415, Route de Gisy, 78943 Vélizy-Villacoublay Cedex, France. E-mail address: [email protected] (A. Launay). 0749-6419/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2011.02.005

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a constitutive law is first needed to describe the local cyclic state, and on the other hand, a fatigue criterion relying on the knowledge of the local variables is designed to predict the fatigue life duration. In this paper, we focus on the former aspect. More specifically, we investigate the mechanical response of the material in order to reach a relevant constitutive law for cyclic loadings. Describing the cyclic behaviour of SGFR thermoplastics is a complex task and raises many difficulties. First, the injection moulding process results in an elaborate microstructure of the composite material. The short glass fibres may not be homogeneously distributed in the polymeric matrix (clusters), their local orientations depends on the moulding flow, and the matrix itself is very sensitive to process conditions (in particular to temperature gradients), resulting in possible inhomogeneities regarding the crystallinity rate. Even if the industrial components are very often shell-like (2D) structures, the physical and mechanical properties of the thermoplastic composite vary across the spatial location, across the section thickness, and depend on the spatial direction (anisotropy). From a mechanical point of view, the highly non-linear behaviour of the polymeric matrix generates other kind of difficulties. To be representative of the actual service life of automotive components, cyclic loadings must involve creep–fatigue coupling and complex histories. Taking into account the fact that no fatigue criterion is generally accepted and used for SGFR thermoplastics, it means that many mechanical values have to be accurately predicted: strain and stress amplitudes, hysteretic areas, cumulative irrecoverable strain, stiffness loss, etc. Last but not least, the temperature and also the humidity rate (in case of a polyamide matrix) greatly affect the mechanical properties of the thermoplastic composite. The industrial components for automotive applications undergo a large range of environmental conditions, which is the reason why the description of the cyclic behaviour has to be valid under and beyond the glass transition temperature. Our objective is to design a constitutive model for the cyclic behaviour compatible with numerical computations in the industrial context. The first stage consists in the simulation of the injection process, so as to take into account the microstructure of the composite material. There exist two ways of integrating the anisotropy in a constitutive model. The first way is often evoked in the literature as homogenization or scale-transition methods. The overall behaviour results from the description of the mechanical response of each phase, and the anisotropy naturally appears as a consequence of the microstructure. The second way is a phenomenological modelling, the material being considered directly from the macroscopic point of view. It may also be supposed as anisotropic, but a relation between the microstructure and the overall mechanical properties must be assumed. These two approaches will be quickly described and compared in the following sections. The scale-transition models aim at building the macroscopic response of the composite material from the interactions between the elastic short glass fibres and the highly non-linear thermoplastic matrix. These methods are grounded on the knowledge of the local microstructure by means of orientation tensors as proposed by Advani and Tucker (1987). In the linear elasticity framework, the determination of the overall response stems from Eshelby (1957)’s results and their extension to the case where the volumetric fraction of the short fibres increases (Benveniste, 1987; Mori and Tanaka, 1973). More recently, several authors have developed homogenization schemes for various fibre orientation distributions (Dray Bensahkoun, 2006; Mlekusch, 1999) and for non-linear (elasto-plastic, viscoelastic, elasto-visco-plastic) mechanical behaviours of the matrix (Doghri and Friebel, 2005; Pierard and Doghri, 2006; Ponte Castañeda et al., 1998). Variants may also be applied in the particular case of a neat semi-crystalline polymer matrix, where both crystalline and amorphous phases are distinguished (Parenteau et al., 2008; Regrain et al., 2009). Even if those micromechanical approaches are physically justified, and can handle complex interactions between fibres and matrix (damage, interfacial debonding, etc.) (Meraghni et al., 2002; Zairi et al., 2008), they require much data to be efficiently used in an industrial context. Moreover, some of this information such as fibre orientation or interfacial behaviour is not always easily available or reliable. Above all, the calculation durations on complex structures with non-linear behaviour are prohibitive. This certainly explains why, to the authors’ knowledge, there is no validation of micromechanical methods on non-linear PMC structures under cyclic loadings in the literature. On the other hand, the phenomenological models are rather easily implemented for structural analysis. Many works regard the non-linear behaviour of semy-crystalline thermoplastics. The constitutive equations developed for these neat polymers are of great interest to understand ways of modelling some typical mechanical features also observed on reinforced thermoplastics. Most of the following works physically rely on the description of deformation micro-mechanisms in semi-crystalline polymers, reviewed by Bowden and Young (1974) then by Lin and Argon (1994). Different mechanical features can be handled by describing the visco-(hyper) elastic response of the macromolecular amorphous network, combined with the viscoplastic effects and damage occuring in the crystalline phase (Détrez, 2008; Kichenin, 1992; Kichenin et al., 1996), as well as in intermolecular amorphous phase (Ayoub et al., 2010). Different viscoplastic models have been used to predict the irrecoverable mechanisms in polymeric materials. One can quote the VBO (viscoplasticity based on overstress) model, originally proposed by Ho and Krempl (2002) and Krempl and Khan (2003) and further modified by Colak (2005) in order to improve the description of the unloading stages, the Bodner–Partom model (Zairi et al., 2005, 2008) or constitutive equations similar to Lemaitre and Chaboche (1985) model with a yield criterion fitted to polymeric materials (Ghorbel, 2008). Other phenomenological models do not associate a deformation mechanism to a physical phase of the composite, but rather to the specific microstructure of a semi-crystalline polymer. Oshmyan et al. (2004) and Hizoum et al. (2006), as well as Drozdov and Gupta (2003), suggest that a semi-crystalline thermoplastic may be considered as a set of mesoregions, bridged by solid links (e.g. crystalline lamellae, chain entanglements) and able to slide along each other. It enables to depict the structural evolution of

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the material (e.g. strain-induced transformation of crystallites, fragmentation of lamellar blocks, relaxation of the amorphous phase). The proposed models accurately describe mechanical features which are specific to polymeric materials: highly nonlinear unloading curves, softening or hardening of the elastic modulus caused by crystallinity variations, high strain recovery at zero-stress. The phenomenological models developed for semi-crystalline polymers can be extended to SGFR thermoplastics, assuming that the above-mentioned deformation mechanisms are similar (Chaboche, 1997; Drozdov et al., 2003, 2005; Oshmyan et al., 2006; Rémond, 2005). In spite of the high constrast between the mechanical properties of the short glass fibres and of the polymeric matrix, the material is considered as homogeneous. However, in the general case (where the fibres are not randomly distributed), the constitutive behaviour of the composite has to be anisotropic. A phenomenological model must therefore include a relationship between the microstructure (for example the orientation tensors) and the overall mechanical properties. This point is challenging because such a relationship requires to test different microstructures (different fibre orientation distributions for example) to be identified, and may involve many parameters. In this paper, we choose to model the mechanical response of the composite with a phenomenological approach, in order to accurately describe non-linear mechanisms in an efficient numerical method for structural computations. We focus on one specific orientation distribution, working on injected tensile specimen. Hence, modelling the anisotropy of the material and taking into account the influence of the fibre orientation distribution on the mechanical response is beyond the scope of the present paper. We rather aim at underlining, analyzing and modelling the complex mechanical effects displayed when loading a thermoplastic composite. This works uses and supplements previous experimental results published by the authors (Launay et al., 2010). Two humidity states have been studied in order to illustrate the relevance of the model under and around the glass transition. Such a work is a necessary step in order to describe cyclic loadings which are encountered during the service life of automotive components (loadings interrupted with pauses, complex loading histories, broad range of environmental conditions). The outline of the paper is as follows. Section 2 deals with a quick presentation of the material of the study and the experimental methods. The experimental results are displayed and analyzed in Section 3, which also includes experimental evidence for complex non-linear mechanisms. The proposed constitutive equations for the cyclic behaviour are introduced in Section 4, along with a methodology for the parameter calibration and the comparison between the numerical predictions and the experimental data. Section 5 discusses the relevancy of the model in comparison with other phenomenological models and raises possible improvements. Final remarks and perspective work are presented in Section 6. 2. Material and experimental techniques The material of this study is a polyamide 66 containing 35 wt% of short glass fibres provided by DuPont de Nemours (DuPont™ ZytelÒ 70G35 HSLX), currently used for automotive applications. Table 1 sums up some of the physical properties of both the semi-crystalline matrix and the short fibres. A SEM fractography displays in Fig. 1 the microstructure of the studied composite. With the help of MOLDFLOW, an injection simulation software, the averaged second order orientation tensor in the working length of the sample has been computed:

2

0:916 0:004 0:000

6 a ¼ 4 0:004 0:000

3

7 0:064 0:000 5 0:000

0:020

ð1Þ ðe1 ;e2 ;e3 Þ

which reveals a highly-oriented orientation distribution along the moulding flow direction. The overall mechanical behaviour is thus expected to be quasi-transversely isotropic, with e1 as main axis. This hypothesis is assessed when computing the e (Mori–Tanaka scheme and orientation averaging). C e is represented with help of Voigt notahomogenized stiffness tensor C tions and can indeed be considered as transversely isotropic:

Table 1 Physical and mechanical properties of the studied PA66GF35. Polyamide 66 (23 °C, RH50) Molecular weight (kg mol1)a

Crystallinity rate (%)a

Melting temperature (°C)a

Young modulus (MPa)

Poisson coef.

34.0

35.6

263

1987

0.40

Glass fibres Volumic fraction (%)

Average fibre length (m)

Average fibre diameter (m)

Young modulus (MPa)

Poisson coef.

19.5

250  106

10  106

72,000

0.22

a These data have been provided by DuPont™. The crystallinity rate and the melting temperature are determined by DSC (Differential Scanning Calorimetry).

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Fig. 1. Fracture surface of a cryogenically broken specimen (SEM), picture size is 150  150 lm (magnification: 300).

Fig. 2. ISO527-2-1A tension specimen (all dimensions are in mm). The arrow stands for the moulding flow direction.

2

13812: 3455: 3360: 40: 0: 6 3455: 5877: 3510: 0: 0: 6 6 6 3360: 3510: 5577: 1: 0: e ¼6 C 6 40: 0: 1: 1176: 0: 6 6 4 0: 0: 0: 0: 1071: 0:

0:

0:

0:

0:

0: 0: 0: 0: 0: 1032:

3 7 7 7 7 7 7 7 7 5

ð2Þ

ð11;22;33;12;13;23Þ

As the polyamide is known to be very sensitive to the relative humidity (RH) of the air, all mechanical tests were processed on the same material but at two humidity states, dry-as-moulded (DAM) material or conditioned at the equilibrium with an air containing 50% of relative humidity (RH50). The tests were not long enough to observe a significant variation of water concentration in the material, which was checked by weighing the samples before and after the tests. Tensile tests were carried out on a servo-hydraulic INSTRON tensile machine (model 1342, in the LBMS laboratory), at room temperature. ISO527-2-1A tensile specimens were injection moulded from this grade, their shape and dimensions are shown in Fig. 2. Local strain was measured by means of an INSTRON contact extensometer (model 2610-601, gauge length 12.5 ± 5 mm), and the tensile load was controlled by a 100 kN capacity load cell. Moreover, the surface temperature of the samples was recorded using an infrared camera (model Phoenix MWIR 9705 by FLIR SYSTEMS). Fig. 3 illustrates the mechanical tests that have been conducted to investigate the cyclic behaviour.

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Fig. 3. Definition of mechanical tests of the experimental study. All tests have been carried out with a tensile load, either stress or strain controlled.

Fig. 4. Temperature sweep at 1 Hz on DAM and RH50 materials (courtesy of DuPont™).

A dynamic mechanical analysis (DMA) has also been performed on both DAM and RH50 materials, in order to study the viscoelastic response at low strains (see Fig. 4). The glass transition temperature of the DAM material is about 80 °C, whereas it decreases to 30 °C for the RH50 material. At room temperature, the conditioned material is thus expected to exhibit the more pronounced viscoelastic effects. This is the reason why we will focus in the sequel on that state of the material, which

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is more complex but nevertheless representative of actual service life. The DAM material always presents the same mechanical features, but in an slighter way. 3. Experimental results and mechanical analysis This section is devoted to the experimental database conducted on the RH50 material at room temperature. The results are analyzed in order to experimentally highlight some mechanical features; this investigative work is the ground for the forthcoming constitutive modelling. 3.1. Static tension tests Static tension tests have been performed on DAM and RH50 materials. The loading rates span four orders of magnitude, from 2.5 to 2500 MPa/s, thus covering the range encountered in the automotive applications. The results are presented in Fig. 5a, for both DAM and RH50 materials. The plasticizing effect of water on the macroscopic response is in evidence: the strain at break increases whereas the stiffness and the ultimate stress decrease. Moreover, the viscoelasticity plays a key role in the mechanical behaviour for the RH50 material, which explains the influence of the stress rate on the stiffness from the very beginning of the loading. On the other hand, the DAM material seems to be stress-rate insensitive while the stress does not overshoot 100 MPa. The evolution of the surface temperature during loading (see Fig. 5b) is also interesting. The same thermoelastic slope for all loading rates is observed, and does not seem to depend on whether the material is dry-as-moulded or conditioned at RH = 50%. The temperature rise, observed on the second stage of the test, can be directly imputed to the conversion of mechanical work into heat (intrinsic dissipation). When the loading rate is high, the dissipative phenomena are hardly visible, and occur just before the final breakdown of the sample. On the contrary, when the loading rate is slow, the temperature increases smoothly until a sudden acceleration when the final fracture propagates through the specimen.

250

Nominal stress (MPa)

increasing stress rate 200 150 2.5 MPa/s DAM 25 MPa/s DAM 250 MPa/s DAM 2500 MPa/s DAM 2.5 MPa/s RH50 25 MPa/s RH50 250 MPa/s RH50 2500 MPa/s RH50

100 50 0 0

1

2

3

4

5

200

250

Nominal strain (%)

(a)

Temperature variation (C)

2

2.5 MPa/s DAM 2.5 MPa/s RH50 25 MPa/s RH50 250 MPa/s RH50 2500 MPa/s RH50

1.5 1 0.5 0 -0.5

increasing stress rate -1 0

50

100 150 Nominal stress (MPa)

(b) Fig. 5. Monotonic tensile tests on DAM and RH50 materials, at various stress rate, until failure. (a) Strain–stress curves and (b) temperature variation versus stress. For the sake of visibility, only one curve is shown for the DAM material in the second graph.

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The influence of the stress rate on the dissipative effects, and the fact that they occur since the very beginning of the loading, are clear hints of short-term viscoelastic effects. The characteristic time scale is about one second. As anticipated with the dynamic mechanical analysis, these short-term viscoelastic effects are much more striking on RH50 material, because at ambient temperature the amorphous phase shows a peak of dissipated energy. 3.2. Cyclic creep-recovery test Monotonic tensile tests cannot be sufficient to investigate non-linear mechanical behaviours. The experimental campaign must include a large set of loading conditions (even only uniaxial) so as to discriminate between many possible modelling solutions, and to better understand the physical micro-mechanisms at stake (Castagnet, 2009; Rémond, 2005). Fig. 6 displays the cyclic creep recovery (CCR) test, which is convenient for studying long-term viscosity effects at several stress levels. At low stresses, time-dependent effects are viscoelastic: creep occurs at the first two stress levels, but the strain is then totally recovered (see Fig. 6b). Let us remark that the viscous scale time is about 100 s, i.e. much longer than the characteristic time underlined on monotonic tensile tests. At higher levels of imposed stress, residual strains appear. At this stage, it is difficult to conclude whether they stem from (visco-) plastic flow or from very long-term recoverable viscosity, as discussed in Pegoretti et al. (2000). 3.3. Anhysteretic curve The anhysteretic curve (see Fig. 7) is also of special interest for the investigation of long-term viscosity effects, because relaxation steps are studied over a large range of imposed strains. Once again, viscous effects occur on a long-term timescale, and appear to be stress- (and not strain-) sensitive: the relaxation amplitudes of the three higher relaxation steps are similar, although the strain levels are increasing. Besides, the mechanical strain may be split in purely elastic and inelastic parts:

160

Nominal stress (MPa)

140 120 100 80 60 40 20 0

Cyclic creep-recovery curve Monotonic tension 25 MPa/s

-20 0

0.5

1

1.5

2

2.5

3

Nominal strain (%)

(a) 140

strain (%) stress (MPa)

120

Nominal strain (%)

2.5

100 2

80

1.5

60 40

1

20 0.5

Nominal stress (MPa)

3

0

0

-20 10

20

30 Time (min)

40

50

(b) Fig. 6. Cyclic creep-recovery test, strain controlled, performed on the RH50 material. Loading and unloading stress rates are 25 MPa/s. (a) Stress–strain curve and (b) stress and strain as functions of the time.

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e¼

r Ee

þ ein

ð3Þ

where Ee is the apparent material stiffness. In Fig. 7b, the inelastic strain rate as a function of the applied stress is plotted for each relaxation step. As the stress decreases during those steps, the inelastic strain ein is considered in a first approximation as essentially viscous. In a linear viscoelastic assumption, the relationship between the applied stress and the viscous strain rate would be a straight line. We thus show that rate-dependent phenomena are highly non-linear. In order to gather all relaxation steps on a single master-curve, the inelastic strain rate is plotted in Fig. 8 as a function of the viscous part of the stress. The latter is defined under the assumption that only a fraction of the total stress activates the viscous mechanisms: 180 160

Nominal stress (MPa)

7

3

120

g

100

2

80 60

f

1

40

e

20

d

0

c

-20 -40

6

5

4

140

0

Anhysteretic curve Monotonic tension 25 MPa/s

b

a 0.5

1

1.5 2 2.5 Nominal strain (%)

3

3.5

4

(a)

Inelastic strain rate (%/s)

0.1

0.05

3

2

4

56

7

1 0 f b -0.1

g

e

-0.05 c

d

a 0

50 100 Nominal stress (MPa)

150

(b) Fig. 7. Anhysteretic curve (defined in Fig. 2), loading and unloading strain rates are 0.25%/s, and relaxation steps last 15 min. (a) Stress–strain curve and (b) analysis of the relaxation steps: inelastic strain rate as function of the applied stress.

Inelastic strain rate (%/s)

0.1

0.05

0

-0.05

-0.1 -100

-50

0 Shifted stress (MPa)

50

100

Fig. 8. Stress shift and attempt to an experimental determination of a non-linear viscous flow rule.

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rv ¼ r  Ev  ein

ð4Þ

Ev (in MPa) stands for a viscous modulus. The corresponding 1D-rheological model is displayed in Fig. 9. The stress shift, resulting from the introduction of the viscous modulus, does not give rise to a single master-curve. The suggested modelling is obviously too simple to describe the non-linear viscous response of the material over the whole range of applied stresses. However, a non-linear viscous flow rule such as:

h

e_ in ¼ A sinh

r im v

ð5Þ

H

is displayed in continuous line in Fig. 8. This thresholdless formulation represents a convenient way of describing the nonlinear viscous effects at moderate or high stress.

Fig. 9. Rheological model used for the analysis of the anhysteretic curve.

125 100 Nominal stress (MPa)

Nominal strain (%)

1.5

1

0.5

strain = 1.75 % 1.5 % 1.25 %

75

1.0 % 0.75 %

50

0.5 % 25 0.2 %

0

0 0

5

10

15

20

25

30

0

5

10

Time (min)

15

20

25

30

Time (min)

(a)

(b)

Fig. 10. TRR tests for different levels of imposed strain: (a) strain versus time and (b) stress versus time. Loading strain rate is 0.2%/s, except for the third curve (emax = 0.75%) for which it is 0.02%/s.

Backstress X (MPa)

75

50

25

0 -0.1

Armstrong-Frederick model data CCR tests data TRR tests data Anhysteretic curve 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Residual strain (%)

1

Fig. 11. Analysis of the TRR tests, the cyclic creep-recovery (CCR) test and of the anhysteretic curve. The hardening back-stress is plotted as a function of the evaluated residual strain.

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3.4. Tension Relaxation Recovery (TRR) tests The TRR tests (see Fig. 10) have been performed on a MTS servo-hydraulic tensile machine, at the LMS laboratory. The relaxation steps can be well correlated with the ones shown on the anhysteretic curve. For low levels of imposed strain, the strain at the end of the recovery step is equal to zero. For higher levels, the recovery time is long enough (up to 10 h) to assess that the observed residual strain is indeed irrecoverable, or at least that the recovering time is long enough to be neglected for the modelling. Residual strains thus appear above a threshold, which can be described in the plasticity framework. For instance, the Armstrong–Frederick kinematic hardening model links the residual ‘‘plastic’’ strain ep to the hardening back-stress X:

X¼

C

c

ð1  ecep Þ

ð6Þ

140

Nominal stress (MPa)

120 100 80 60 40 20 0

Cyclic tension-tension test, 2.5 MPa/s Monotonic tension 2.5 MPa/s

-20 0

0.5

1 1.5 Nominal strain (%)

2

2.5

(a) 3 cycles per stress level (50, 75, 100, 125 MPa) 200

Nominal stress (MPa)

180 160 140 120 100 80 60 40 20 Cyclic tension-tension test, 250 MPa/s Monotonic tension 250 MPa/s

0 -20 0

0.5

1

1.5 2 Nominal strain (%)

2.5

3

(b) 3 cycles per stress level (40, 70, 95, 120, 140 MPa) 160

Nominal stress (MPa)

140 120 100 80 60 40 20 0 Cyclic tension-tension test, 0.016%/s Monotonic tension 2.5 MPa/s

-20 -40 0

0.5

1

1.5

2

2.5

Nominal strain (%)

(c) 3 cycles per strain level (0.83, 1.7, 2.7 %) Fig. 12. Cyclic tensile–tensile tests, stress or strain controlled. (a) Low stress rate: 2.5 MPa/s, (b) high stress rate: 250 MPa/s and (c) low strain rate 0.016 %/s. The monotonic tension tests corresponding to the imposed stress rates are shown.

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In uniaxial tensile loading, X is the difference between the applied stress r and the plastic threshold r0. C (MPa), c (–) and r0 (MPa) are three model coefficients, determined by a least-square fitting technique. X = rrelaxed  r0 is built from the relaxed stress (measured at the end of relaxation steps, see Fig. 10b), while ep is measured at the end of the recovery stage (see Fig. 10a), and each TRR test provides a data point in Fig. 11. Data points are extracted similarly from the anhysteretic curve and from the cyclic creep recovery test. In the latter case, the relaxed stress is equal to the imposed stress during each creep step, but the ‘‘plastic’’ strain (measured after recovery) might be less reliable due to the fact that recovery steps are shorter. However, we show that an Armstrong–Frederick hardening law fairly fits the data, in spite of the experimental scattering. Especially, the matching is far better than in the case when we use the maximal stress in the definition of X. Eq. (6) can therefore be used for the prediction of the residual strain evolution under cyclic loading. 3.5. Cyclic tensile–tensile tests 3.5.1. Hysteretic loops and residual strain The investigation of the cyclic behaviour is deepened by the mean of cyclic tensile–tensile tests performed at two different stress rates (see Fig. 12a and b). A third analogous but strain-controlled test has also been carried out (see Fig. 12c). Viscous effects explain why in the latter case compression steps are observed; they remain quite moderate in order to avoid sample buckling. The comparison between the two stress-controlled tests shows that hysteretic loops are bigger at low stress rate, which is consistent with the classical viscoelastic models. Short-term viscosity should be held responsible for that phenomenon. Moreover, considering that residual strains (at the end of the unloading steps) are much more important at 2.5 MPa/s than at 250 MPa/s for equivalent imposed stresses, we may also suggest that the irreversible flow (and not only the recoverable viscous flow) is also strain-rate dependent, and then should be called ‘‘viscoplastic’’ rather than ‘‘plastic’’. 3.5.2. Cyclic softening Cyclic stiffness loss is observed on cyclic tensile–tensile tests in Fig. 12. In order to emphasize it, the apparent stiffness E is measured at the beginning of each loading step. The normalized stiffness loss EE0 is then computed. Note that this quantity is

Fig. 13. Analysis of the apparent stiffness loss as a function of (a) the inelastic energy density or (b) the inelastic strain. The stiffness is normalized by the initial stiffness, measured at the beginning of each test.

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independent of strain-rate effects, as the initial stiffness E0 is calculated at the same strain- (or stress-) rate as the current one E. In Fig. 13a, the normalized stiffness loss is plotted as a function of the inelastic density of energy Win. This energy is defined as the difference between the total mechanical energy and the (instantaneous) elastic strain energy:

W in ¼ W tot  W el ¼

Z

eðtÞ

r de  0

r2 2E

in the uniaxial case

ð7Þ

All tensile–tensile cyclic tests as well as the TRR tests and the anhysteretic curve have been analyzed. Unlike Détrez (2008), the correlation between the cyclic softening (evaluated through the apparent stiffness decrease) and the residual strain er has been found very poor (see Fig. 13b), whereas it is far better with Win. As a matter of fact, even if the correlation is not perfect (partly because of experimental scattering), the cyclic softening appears to be independent of the type of loading, which is not the case if the correlation is made along with er. As presented in Fig. 13a (in bold continuous line), we suggest a phenomenological law accounting for the apparent stiffness evolution along with the inelastic energy density:

   W in E ¼ E0 1  a 1  e b

ð8Þ

where a is the maximal relative stiffness loss and b a characteristic energy density. The cyclic softening could involve many physical sources (transformation of the semi-crystalline matrix structure, fibre/ matrix debonding, void formation in the bulk matrix or at fibre ends) as mentioned in Lin and Argon (1994) and Oshmyan et al. (2004). SEM or optical microscopy should be conducted to correlate micro- and macro-physical phenomena. It could help us to assess if the softening stems from irreversible phenomenon such as cyclic damage, or if it rather results from viscoelastic effects, reversible on a (very) long time scale. 4. Modelling the cyclic behaviour 4.1. Constitutive equations The experimental study exhibits several mechanical features that must be described by the phenomenological model:     

recoverable viscoelastic effects occur on both short and long time scales; at high stress levels, a non-linear viscous flow law is required; evidence has been given for irreversible strain above a stress threshold; the Armstrong–Frederick kinematic hardening law appears to fit well the residual strain evolution; cyclic softening has been successfully correlated to the inelastic energy density.

This work involves uniaxial tensile tests performed on one specific orientation distribution. It thus cannot be sufficient in order to characterize a 3D anisotropic constitutive behaviour of the composite material, which is beyond the scope of this paper. Nevertheless, the microstructure of the studied material requires to write the constitutive equations in a 3D anisotropic frame, which takes into account all physical mechanisms at stake in the material. In a second step, the model is reduced to an unidimensional form, which is consistent with our experimental database, and enables the identification of some of its parameters. 4.1.1. General 3D formulation The proposed model lies within the framework of the generalized standard materials (GSM) defined by Halphen and Nguyen (1975). Five tensorial and one scalar state variables are needed to define the current mechanical state, as well as their associated thermodynamic forces:      

the the the the the the

overall strain e, outside control variable, associated to r; long-term viscoelastic strain ev1, associated to Av 1 ; short-term viscoelastic strain ev2, associated to Av 2 ; pseudo-viscoplastic strain evp, associated to Av p ; hardening variable a, associated to the centre of the pseudo-viscoplastic surface X; softening variable, called b, and associated to Ab .

The overall strain is split into:

e ¼ ee þ ev 1 þ ev 2 þ ev p

ð9Þ

The elastic strain we introduce here is not an internal variable, but is convenient for the equation writing. Besides, for the numerical implementation of the law, we thus choose to work with ee and p, the cumulative viscoplastic strain, and no longer with evp.

A. Launay et al. / International Journal of Plasticity 27 (2011) 1267–1293

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At the reference temperature T0, the Helmholtz free energy is written as the sum of the stored energies, and depends on the state variables:





1 2

1 2

1 3

q0 w0 e; ev 1 ; ev 2 ; ev p ; a; b ¼ ev 1 : Cv 1 : ev 1 þ ev 2 : Cv 2 : ev 2 þ C a : a þ

   1 e  ev 1  ev 2  ev p : Ce ðbÞ : e  ev 1  ev 2  ev p 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð10Þ

ee

The thermodynamic forces are the partial derivatives of the volumic free energy along the corresponding state variable. We thus define the state equations of the problem:

r¼q

  @w0 ¼ Ce ðbÞ : e  ev 1  ev 2  ev p @e |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð11Þ

ee

Av 1 ¼ q

@w0 ¼ r  Cv 1 : ev 1 @ ev 1

ð12Þ

Av 2 ¼ q

@w0 ¼ r  Cv 2 : ev 2 @ ev 2

ð13Þ

Av p ¼ q

@w0 ¼r @ ev p

ð14Þ

@w0 2C ¼ a 3 @a

ð15Þ

X ¼ q

Ab ¼ q

@w0 1 @Ce ¼  ee : : ee 2 @b @b

ð16Þ

i

where C stands for the fourth order elastic tensor, used for the instantaneous elasticity or for the two viscoelastic Maxwell elements. At this stage, no further consideration is made regarding their 3D form: fully anisotropic, orthotropic, isotropic, etc. The softening depends on the variable b according to the following equations:

   b Ce ðbÞ ¼ gðbÞ Ce ð0Þ with gðbÞ ¼ 1  a 1  exp b

ð17Þ

Note that on a theoretical point of view, the cyclic softening may influence other material properties than the stiffness tensor, such as viscoplastic parameters for example. This could stand for an extension of the proposed modelling, but would require specific tests for an experimental justification. Moreover, previous experimental works published by Noda et al. (2001) and Mourglia Seignobos (2009) show that fatigue loading on SGFR polyamide indeed results in Young modulus decrease, whereas other macroscopic parameters (such as the loss factor tan d or the dissipated energy Wdiss) remain constant. The proposed model for the cyclic softening is in accordance with these observations. A second potential, convex and minimum in 0, is required in order to achieve the description of the behaviour. According to the GSM theory, we define the dual dissipation potential u⁄, depending on thermodynamic forces as well as state variables1:

u ¼ u ðr; Av 1 ; Av 2 ; Av p ; X; Ab ; e; ev 1 ; ev 2 ; ev p ; a; bÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} state variables

13m 0  Z f r; X; a 3 devðAv 1 Þ : devðAv 1 Þ 3 devðAv 2 Þ : devðAv 2 Þ 4sinh @ A5 df þ ~  ðAb ; state variablesÞ þ þu ¼ H 4g1 4g2 A 2

ð18Þ

As in viscoplastic models, f is the yield function. Following Germain et al. (1983) and Chaboche (1997), we suggest a formulation abiding by the GSM framework:

f





r; X; a ¼ J v p þ

  3c 4 X : X  C2a : a 9 4C

ð19Þ

J v p is the equivalent stress built on the tensor Av p centered on X:

J vp ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðAv p  XÞ : P : ðAv p  XÞ

ð20Þ

1 u⁄ is a differentiable function. Observing that f P 0, we demonstrate that its second derivative is always positive. Moreover, f(0) = 0 and g0 (b) 6 0 (implying Ab P 0) explain why u⁄ is minimal in 0.

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P is a symmetric fourth order tensor. In the isotropic case, it is equal to 32 K, which leads to the equivalent von Mises isotropic equivalent stress.2 In a more general way, this tensor describes an anisotropic equivalent stress. For example, Hill (1948)’s orthotropic criterion or generalized versions (Oller et al., 2003; Voyiadjis and Thiagarajan, 1995) are widely used for anisotropic elasto-viscoplastic materials. Besides, Barlat et al. (1997, 2005) suggests different kinds of criteria based on the eigenvalues of a linear-transformed stress, the transformation being similar to Hill’s orthotropic tensor. P may be pressure insensitive, which reads Pijkk = 0, or may describe an unsymmetry between tension and compression in different material directions (Tsai and Wu, 1971). ~  is the part of the dissipative potential related to the cyclic softening. It must describe its evolution according to an u inelastic energy. In order to model the fact that the softening is negligible at low stress levels, only the viscoplastic part of the dissipated energy is assumed to activate the softening. We thus postulate an expression of that potential as:



~  ðAb ; state variablesÞ ¼ Ab  e_ : Ce ðbÞ : ðe  ev þ ev þ ev p Þ  u vp 1 2

2C a_ : a 3

ð21Þ

~  in which the rates of the state variables e_ and a_ are Eq. (21) is a condensed formulation of the rigourous expression of u vp written as functions of the state variables themselves (combining Eqs. (24) and (25) with Eqs. (11)–(16)). The evolution laws of the mechanical problem are written accordingly to the GSM theory:

e_ v 1 ¼

@ u 3 ¼ devðAv 1 Þ @Av 1 2g1

ð22Þ

e_ v 2 ¼

@ u 3 ¼ devðAv 2 Þ @Av 2 2g2

ð23Þ

e_ v p ¼

   m P : ðAv p  XÞ @ u @ u @f J vp ¼ ¼ A sinh  @Av p @f @ r H J vp |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

ð24Þ

kv p P0

a_ ¼

nv p

@ u @ u @f ¼ ¼ kv p ðnv p þ caÞ @X @f @X

ð25Þ

@u 2C b_ ¼ ¼ e_ v p : Ce ðbÞ : ee  a_ : a 3 @Ab 

ð26Þ

kvp is the (pseudo-) viscoplastic multiplier, and nvp is the direction that maximizes the pseudo-viscoplastic dissipation. In the present form, the viscoelastic as well as the pseudo-viscoplastic strains are isochoric, which reads Tr(ev1) = Tr(ev2) = Tr(evp) = Tr(a) = 0. At last, the instantaneous dissipated energy is calculated as the sum of the rates of each state variable multiplied by the associated thermodynamic force:

3 3 D ¼ Av 1 : e_ v 1 þ Av 2 : e_ v 2 þ Av p : e_ v p þ X : a_ þ Ab  b_ ¼ Av 1 : K : Av 1 þ Av 2 : K : Av 2 2g 2g 1

kv p 2 1 þ ðAv p  XÞ : P : ðAv p  XÞ þ kv p C ca : a  g 0 ðbÞ ee : Ce ð0Þ : ee 3 2 |ffl{zffl} J vp

2

ð27Þ

60

Using the fact that Ce , K and P are symmetric positive tensors, and the positivity of kvp, it is obvious that D is always positive, as required by the second principle of thermodynamics. Moreover, re-writing Eq. (26) leads to:

b_ ¼ e_ v p : Av p þ a_ : X

ð28Þ

which shows that b has the physical meaning of a cumulative energy dissipated by the pseudo-viscoplastic phenomena, which is consistent with our modelling. Remark. We call evp the pseudo-viscoplastic part of the overall strain, and not merely the viscoplastic part. A classical viscoplastic formulation indeed exhibits a plastic threshold, or a plastic surface, defined by f = 0. In our modelling without threshold, the yield function f is always positive, but the choice of the dissipative potential (a hyperbolic sine with power m) explains that at low stress levels, the rate e_ v p is close to zero. The proposed constitutive law is thus a thresholdless viscoplastic formulation, called pseudo-viscoplastic.

2 A spectral decomposition I ¼ J þ K is classically suggested. I is the fourth order identity tensor on the symmetric second order tensors, and J ¼ 13 1  1 is the projector on the spherical part. K is the projector on the deviatoric part, since K : t ¼ t  13 ð1 : tÞ1 ¼ devðtÞ.

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4.1.2. 1D formulation of the constitutive model The experimental tests are conducted on tensile specimen. The stress tensor is therefore unidimensional: r = r(t)e1  e1. Under the assumption of a transversely isotropic equivalent stress J v p , which has been proved to be a reasonable hypothesis regarding the specific orientation distribution for the ISO527-2-1A specimen, it can be shown that the backstress X and the hardening variable a are diagonal and only depend on a single scalar:

XðtÞ ðe2  e2 þ e3  e3 Þ 2 aðtÞ ðe2  e2 þ e3  e3 Þ a ¼ aðtÞe1  e1  2 X ¼ XðtÞe1  e1 

In the transversely isotropic case around e1axis, using Voigt’s notation and classical Hill’s orthotropic representation, P is represented as:

2

2B B B 0 0 6 B ðB þ DÞ D 0 0 6 6 6 B D ðB þ DÞ 0 0 P¼6 6 0 0 0 2E 0 6 6 4 0 0 0 0 2E 0

0

0

0

0

0

3

0 7 7 7 0 7 7 0 7 7 7 0 5 2G

ð29Þ

ð11;22;33;12;13;23Þ

In the same manner, if the linear viscoelastic tensors are assumed to be transversely isotropic (or even isotropic, as suggested by Andriyana et al. (2010)), the long- and short-timescale viscoelastic strains are written:

ev 1 ¼ ev 1 ðtÞe1  e1 

ev 1 ðtÞ

ðe2  e2 þ e3  e3 Þ 2 e ðtÞ ev 2 ¼ ev 2 ðtÞe1  e1  v 2 ðe2  e2 þ e3  e3 Þ 2 No particular hypothesis is made regarding the softenable elastic tensor Ce . The 3D model reduces then to a 1D formulation, as follows:

r ¼ Ee ðbÞðe  ev 1  ev 2  ev p Þ ¼ Ee ðbÞee

ð30Þ

2C a X¼ 3 1 e_ v 1 ¼ ðr  Ev 1 v 1 Þ

ð31Þ ð32Þ

g1

e_ v 2 ¼

1

g2

ðr  Ev 2 v 2 Þ

ð33Þ

    m jr  3X j 3X 2 sign r  e 2 H      3X  m  j r  j 3X 2 ~a þc a_ ¼  Ae sinh sign r  e 2 H 3X b_ ¼ re_ v p þ a_ 2 Ee ðbÞ ¼ gðbÞE0e 

e_ v p ¼ Ae sinh

ð34Þ ð35Þ ð36Þ ð37Þ 1

The elastic and viscoelastic moduli Ei are defined as follows. Introducing Si ¼ Ci , its first component Si1111 is written pffiffiffiffiffiffi pffiffiffiffiffiffi 1/Ei. e e Moreover, pffiffiffiffiffiffi in the case of a transversely isotropic pseudo-viscoplastic yield function, we use A ¼ A 2B, H ¼ H= 2B and c~ ¼ c= 2B. The 1D-rheological scheme is pictured in Fig. 14. The different parameters as well as their physical meaning are summed up in Table 2. Note that the identified parameters are valid for one specific microstructure, characterized by the given orientation distribution tensor of the ISO527-2-1A specimen (see Section 2). 4.2. Identification of the model parameters The identification process is an important part of the modelling. Such a complex constitutive model has indeed to be completed with an specific identification methodology. The first reason is pragmatic: because of the high number of parameters, an optimization algorithm cannot find the optimal set of parameters by its own means and might encounter difficulties to converge toward an absolute minimum in a 12-dimension space. On a more fundamental point of view, the optimization strategy aims at uncoupling at best all mechanical features in order to ensure that all parameter values have a physical meaning. We here depict this strategy and thus identify all parameters step-by-step.

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Table 2 Coefficients involved in the proposed constitutive model. The numerical values are given for the dry-as-moulded (DAM) and for the conditioned (RH50) material at room temperature. Mechanical feature

Model parameter

Symbol

Unit

DAM

RH50

Equation

Elasticity

Initial Young modulus

E0e

MPa

1.25  104

1.13  104

(17)

Long-term viscoelasticity

Stiffness modulus Viscosity Characteristic time

Ev1

g1 s1

MPa MPa s s

2.25  105 1.87  107 82.9

6.27  104 3.36  106 53.6

(12) (12) s1 = g1/Ev1

Short-term viscoelasticity

Non-linear hardening Non-linear viscosity

Cyclic softening

Stiffness modulus Viscosity Characteristic time

Ev2

g2 s2

MPa MPa s s

9.64  105 2.15  106 2.23

3.39  104 1.24  104 0.34

(13) (13) s2 = g2/Ev2

Hardening modulus Non-linear parameter

C c~

MPa –

6.21  104 6.87  102

3.61  104 5.45  102

(15) and (19) (19)

Characteristic rate

e A e H

s1

2.41  1011

3.29  107

(18) and (24)

Characteristic stress

MPa

17.1

46.7

(18) and (24)

Exponent

m

–

2.72

7.44

(18) and (24)

Maximal softening Charact. energy density

a b

% J m3

7.13 2.24  105

25.0 1.10  105

(17) (17)

Fig. 14. 1D-rheological scheme of the proposed constitutive model.

The identification is processed with the help of ZEBULON (ZeBuLoN, 2005), a FEM software developed at the École Nationale Supérieure des Mines de Paris and ONERA,3 a french aerospace research institute, which includes a powerful and convenient optimization algorithm, based on a Levenberg–Marquardt technique. The optimized sets of parameters for both dry-asmoulded and conditioned materials are given in Table 2.

4.2.1. Short-term viscoelasticity We first determine the influence of the loading rate on the initial stiffness. We should work at sufficient low stress so that pseudo-viscoplastic mechanisms may be neglected, and by consequence no softening occurs either. The initial part of the monotonic tensile tests last short time (even at 2.5 MPa/s) so that long-term viscoelastic effects do not have to be modelled. The proposed model is thus equivalent to a first-order Kelvin–Voigt model. Fig. 15 demonstrates the relevance of our approach, and we are thus able to identify E0e , Ev2 and g2 (see Fig. 14 for the parameters definition). The good accuracy of the response shows that a linear viscoelastic model is sufficient to accurately predict the rate-sensitive initial stiffness over four loading-rate decades. Besides, the ratio Egv22 , which stands for a viscous characteristic time s2, is around one second. This value seems reasonable and is in agreement with the expression ‘‘shortterm’’ viscoelasticity.

4.2.2. Long-term viscoelasticity Once the short-term viscoelasticity has been characterized, we focus on the first step of the cyclic creep-recovery test in order to study the time-dependent response of the mechanical behaviour at long-term time-scale. The stress is low enough to neglect the pseudo-viscoplasticity as well as the softening: the proposed model may be summed up in that case by a second-order Kelvin–Voigt model. 3

Office NAtional d’Etudes et de Recherche Aérospatiale.

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30

Stress (MPa)

25 20 15

Present model Present model Present model Present model 2.5 MPa/s, test 25 MPa/s, test 250 MPa/s, test 2500 MPa/s, test

10 5 0

0

0.05

0.1

0.15 0.2 Strain (%)

0.25

0.3

0.35

Fig. 15. Identification of the short-time scale viscoelastic parameters on the beginning of the monotonic tensile tests.

30

0.35 0.3

25

0.25 Strain (%)

Stress (MPa)

20 15 10 5

0.1 0.05

0 -5 -0.05

0.2 0.15

0

Present model CRR test, 25 MPa/s 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.05

Present model CRR test, 25 MPa/s 0

100

200

300

Strain (%)

Strain (%)

(a)

(b)

400

500

600

Fig. 16. Identification of the long-time scale viscoelastic parameters on the first step of the CCR test. The use of the recorded input signal (and not of the theoretical set point) for the simulation explains the slightly shaky response during creep or strain recovery.

Fig. 16 displays the result of the identification process on both Ev1 and g1 parameters. Creep and recovery rates or amplitudes are described with good accuracy, and we checked that for the TRR tests at low imposed strain (those with entire recoverability of the strain) the prediction is consistent too. Moreover, the identified set of viscoelastic parameters is physically justified. The inequality s1  s2 is verified and two different time scales are well taken into account. 4.2.3. Non-linear pseudo-viscoplasticity e H, e m) and two for the non-linear Five parameters have to be determined: three for the non-linear viscous flow rule ( A, ~ kinematic hardening law (C and c). As the anhysteretic curve has particularly driven us into the present modelling, we use the loading steps of this test to identify these five parameters. In order to further the optimization process, the starting set of parameters has to be as relevant as possible. The coefficients found during the mechanical analysis presented in Sections 3.2, 3.3 and 3.4 have thus been used as starting point. The viscoelastic parameters involved in the model are those previously determined. At this stage of the identification procedure, the softening is inactivated, by setting a = 0. This assumption is justified by the fact that the maximal softening for this test remains moderate. The least-square fitting technique consists in minimizing the difference between the experimental and the simulated stress versus time evolutions. The result is shown in Fig. 17 and clearly demonstrates the usefulness of our procedure. As a matter of fact, relaxation rates as well as relaxation amplitudes are very well described for the whole loading stage. Moreover, many other models often failed to simulate the anhysteretic curve, which persuades us of the interest of the proposed viscous flow law. 4.2.4. Softening parameters In Section 3.5.2 a phenomenological softening model has been suggested in order to take into account the stiffness decreasing along with the dissipated energy. The two parameters a and b have been optimized so that the monotonic tensile tests are described at best. We also made a slight modification on the pseudo-viscoplastic parameters, sticking to this guideline.

A. Launay et al. / International Journal of Plasticity 27 (2011) 1267–1293

160

160

140

140

120

120 Stress (MPa)

Stress (MPa)

1284

100 80 60 40

100 80 60 40

20

20

Present model Anhysteretic curve, test

0 0

0.5

1

1.5

2 2.5 Strain (%)

3

Present model Anhysteretic curve, test

0

3.5

4

4.5

0

500 1000 1500 2000 2500 3000 3500 4000 4500 Strain (%)

(b)

(a)

Fig. 17. Identification of the non-linear pseudo-viscoplastic parameters on the loading part of the anhysteretic curve. The cyclic softening is here inactivated.

200

200 150

2.5 MPa/s, test Present model 25 MPa/s, test Present model 250 MPa/s, test Present model 2500 MPa/s, test Present model

100

50

0

Stress (MPa)

Stress (MPa)

150

0

1

2 Strain(%)

3

100 50 0 −50 −100

4

All steps Present model 0

1

(a)

2 Strain (%)

3

4

(b)

e H, e m, C Fig. 18. Compromise between the monotonic tensile tests and the anhysteretic curve in order to identify the parameters a and b. The coefficients A, ~ are slightly modified. and c

140

3

120

2.5 2 Strain (%)

Stress (MPa)

100

Step 25 MPa Step 50 MPa Step 75 MPa Step 100 MPa Step 125 MPa Present model

80 60

Step 25 MPa Step 50 MPa Step 75 MPa Step 100 MPa Step 125 MPa Present model

40 20

1.5 1 0.5

0

0

0

0.5

1

1.5 Strain (%)

(a)

2

2.5

3

0

500

1000

1500 2000 Time (s)

2500

3000

3500

(b)

Fig. 19. Validation of the present model on the CCR test. (a) Comparison of the numerical results with (continuous line) or without (dashed line) modelling the cyclic softening. (b) Stress versus time curve.

The final compromise we reached is presented in Fig. 18. We notice that the results on the anhysteretic curve are a little less accurate, but the prediction of the relaxation steps remains satisfying. 4.3. Validation on the RH50 material The following figures display, on the tests which are not part of the identification process, the comparison between the prediction of the proposed model and the experimental data. The identified set of parameters has been used. The stress levels

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120

Stress (MPa)

100 80 60 40 20 0 0

0.2

0.4

0.6

0.8 1 Strain (%)

1.2

1.4

1.6

1.8

(a) Stress-strain curve

120 1.5

80

Strain (%)

Stress (MPa)

100

60 40

1

0.5

20 0 0

200

400

600

800

1000

0 800

900

1000

1100

1200

1300

Time (s)

Time (s)

(b) Stress relaxation

(c) Strain recovery

1400

1500

Fig. 20. Validation of the present model on the TRR tests.

reach 150 MPa, which covers the whole range of fatigue loadings. Beyond this value, the constitutive model appears to be less relevant. The comparison between numerical prediction and experimental data for the whole CRR test (the two first steps have used for the determination of the long-term viscoelasticity) is plotted in Fig. 19. At higher stress, where pseudo-viscoplastic mechanisms are activated, the prediction is still reliable, and creep or recovery rates are fairly described. Moreover, the comparison between the mechanical responses with Ee ¼ E0e ¼ const instead of Eq. (17) clearly demonstrates the purpose of the softening model (see Fig. 19a). The matching between numerical simulation and experimental data on TRR tests is displayed in Fig. 20. The stress relaxation is predicted with good accuracy for all imposed strain levels, even when irreversible mechanisms are activated. The strain recovery rates and amplitudes are also well described, but our model tends to over-estimate the residual strains for the last three tests (eimposed > 1%). In the same way as explained above, the main reason essentially lies in the fact that the end of the unloading steps are too linear in our predictions, whereas it seems that non-linear mechanisms are physically activated. In spite of this lack in the modelling, the TRR tests are successfully simulated, and we should remark that the transition from a viscoelastic recoverable response to a viscoplastic irrecoverable response is well depicted by the proposed model. The results regarding the cyclic tensile–tensile tests are shown in Fig. 21. Despite the above-mentioned over-estimation of the tangent stiffness at the end of the unloading steps (for the higher stress levels), the fitting between the model prediction and the experimental data is good. For the stress-controlled tests, the maximal strains are well simulated, so is the viscoplastic ratchetting observed between the first cycle and the two other ones. For the strain-controlled test, the observations are analogous: maximal stresses and cyclic stress decreasing are well predicted. Let us notice that the cyclic softening is also fairly described. 4.4. Validation on the DAM material The proposed model has also been applied to the DAM material, for which a similar experimental campaign has been conducted. The identification process is exactly the same as for the RH50 material.

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140 120

Stress (MPa)

100 80 60 Step 50 MPa Step 75 MPa Step 100 MPa Step 125 MPa Present model

40 20 0 0

0.5

1

1.5 Strain (%)

2

2.5

(a) Cyclic tensile-tensile test, 2.5 MPa/s

Stress (MPa)

150

100

Step 40 MPa Step 70 MPa Step 95 MPa Step 120 MPa Step 140 MPa Present model

50

0 0

0.5

1

1.5 Strain (%)

2

2.5

(b) Cyclic tensile-tensile test, 250 MPa/s 150

Stress (MPa)

100

50

0

Step 0.83 % Step 1.7 % Step 2.7 % Present model

−50 0

0.5

1

1.5 Strain (%)

2

2.5

3

(c) Cyclic tensile-tensile test, 0.016 %/s Fig. 21. Validation of the present model on the cyclic tensile–tensile tests.

We display in Fig. 22 the monotonic tensile tests. As expected, we notice that the overall mechanical response is easier to simulate, notably because the initial stiffness is nearly insensitive to the loading rate. The optimized set of parameters shows indeed that the value of E2 is far greater than E0e or E1, which means that the second viscoelastic element (with short characteristic time) plays no role for the DAM material. On the other hand, the viscoplasticity of the material is less pronounced, even if very well predicted by the proposed model over the whole range of stress rates. In Fig. 23 are shown the comparison between the model prediction and the experimental cyclic curves. The results are in good agreement, and as mentioned for the RH50 material, the main shortcoming of our modelling lies in the under-estimation of the strain recovering during the unloading steps. But in spite of this reservation, we show that the viscoelastic creep

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300

Stress (MPa)

250 200 2.5 MPa/s, test Present model 25 MPa/s, test Present model 250 MPa/s, test Present model 2500 MPa/s, test Present model

150 100 50 0

0

0.5

1

1.5 2 Strain(%)

2.5

3

3.5

Fig. 22. Validation of the present model in different loading cases, for the DAM material.

2.5

200

Step 25 MPa Step 50 MPa Step 75 MPa Step 100 MPa Step 125 MPa Step 150 MPa Step 175 MPa Present model

1.5

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and recovery steps are well described (Fig. 23a and b). At higher loading levels, the non-linear viscous flow is accurately predicted, as seen in Fig. 23b and also in Fig. 23c where the stress relaxation amplitudes are very well simulated. Eventually, the matching between the numerical prediction and the experimental data on cyclic tensile–tensile tests is convincing: the cyclic hardening and the maximal strains (or stresses if the test is strain-controlled) are in good agreement with the measured values. These results strengthen our idea that the constitutive equations developed for the RH50 material are still relevant for the DAM case. Moreover, the mechanical features being less pronounced below the glass transition temperature, it explains why our modelling gives rather accurate results until 200 MPa. Such stress levels are higher than the typical loadings encountered for fatigue applications, but illustrate the predictivity of the proposed constitutive model. 5. Discussion 5.1. Comparison with other phenomenological models We here discuss a few alternative ways of modelling the cyclic behaviour and explain why these ideas have caught our attention but have not finally been chosen. In Fig. 24, we present the rheological schemes associated to each of those ideas. The two-layers model has been proposed by Kichenin (1992) and Kichenin et al. (1996), who observed on a neat polyethylene matrix uncoupled viscous and plastic mechanisms, in the same way we noted it for our material. Kichenin originally used a linear kinematic hardening law, but we tested a generalized version of his model, with non-linear kinematic hardening. The viscous flow rule is non-linear, and stems from a Norton–Hoff viscous potential:

e_ v ¼

 n 3 J v K : rv 2 K Jv

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where ev is the strain of the non-linear viscous element, and J v the equivalent von Mises stress related to the stress in the viscous layer, rv. Such a model, in spite of its simplicity and its interesting features, is not satisfying. The monotonic tensile tests cannot indeed be well described at all stress rates (see Fig. 25a). Such parameters, for which the initial stiffness is well described, lead to non-existent long-term viscoelastic features. A non-linear modelling of the viscous effects cannot be sufficient to represent all the complexity of viscous effects on different time scales. In order to improve the description of the short- and long-term viscous effects, we can picture the EP-KV2 model which includes two viscoelastic time scales (see Fig. 24a). But a mere application of that model on the monotonic tensile tests proves that a rate-independent plasticity is not suitable in order to describe the experimental response (see Fig. 25b). Incidentally, one can notice that the viscoplastic formulation of the constitutive law, with a Norton flow rule as proposed by Chaboche (1997), or with a hyperbolic sinus flow rule as suggested by Szmytka et al. (2010), is very convenient for the description of the non-linear parts, as shown in Fig. 25c. The initial stiffness in the case of the EVP model presents no sensitivity to the stress rate, and no long-term viscoelasticity either. That is the reason why it should be completed by a second order Kelvin–Voigt model, which describes those two features. We can indeed observe, looking at Fig. 25d, that the EVP-KV2 model describes with high accuracy the monotonic tensile tests, over the whole stress-rates range. Moreover, the long-term viscoelastic element allows a good prediction of the creep or recovery steps at low stresses. Such a model may thus seem an interesting idea for the description of the cyclic behaviour of the studied material.

Fig. 24. Rheological schemes of other constitutive models.

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However, as we mentioned before, the anhysteretic curve is an excellent test in order to discriminate several ways of modelling the cyclic behaviour, and especially its viscous features at moderate or high stresses. Fig. 26 displays the comparison between the prediction of this test with the EVP-KV2 model and the proposed model. In spite of the compromise we mentioned in the previous subsection, it is obvious that especially the loading stage is described with more accuracy thanks to our modelling. When plotting the stress as a function of the time, one can notice that the viscous rates during the relaxation steps are much closer to the physical response of the material. That is the consequence of the choice of the non-linear viscous flow law we based upon the analysis of the anhysteretic curve (see Section 3.3). 5.2. Possible improvements of the model The main shortcoming of the model is its inability to accurately describe the unloading paths when high stresses have been reached: the predicted mechanical response seems to be too stiff at the end of those stages, whereas the experimental curve let us think that a viscous flow is activated. Referring to the literature on that specific topic for polymeric materials, we are under the impression that this problem is a difficult issue for many constitutive models (Rémond, 2005). The highly non-linear unloading curve could be modelled by an asymmetry of the yield function, depending on the loading or unloading nature of the applied mechanical signal. That asymmetry of the yield function is usually a consequence of the dependency on the hydrostatic stress (Ghorbel, 2008), but in our case, it would require a criterion leading to a yield function lower for compressive states (combined with a kinematic hardening), which is contrary to the physical observations. Another way, following Colak (2005) and Ayoub et al. (2010), would consist in changing the elastic stiffness while loading or unloading. In our case, such an idea is not convenient, because the initial stiffness at the beginning of the unloading stages is well predicted: it seems more interesting to modify either the yield function f, or the flow rule (Eq. (24)). The latter guideline would catch up with Drozdov and Christiansen (2003), who study the tensile behaviour of a neat polyethylene matrix at small strains. The authors explain the highly non-linear unloading curves by the activation of specific irreversible mechanism during retraction. The introduction of microstructural elements (Oshmyan et al., 2006) would also consist in another way of modelling in order to accurately describe these unloading paths. Eventually, Parenteau et al. (2008) focus on a neat semi-crystalline polypropylene, and observe a specifically non-linear behaviour while unloading. The authors add an exponential factor depending on the hydrostatic stress in order to model this phenomenon. Cyclic ratchetting or stress relaxation are also slightly over-predicted by the proposed model when high stress (or strain) levels are achieved. This drawback has already been observed for metallic materials and has been attributed to the

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Armstrong–Frederick modelling of the kinematic hardening evolution, as discussed by Chaboche (1991) and Ohno and Wang (1993). Alternative constitutive modelling have been suggested in the literature (Kang, 2008), and could also be considered for SGFR thermoplastics. Several modelling ways are therefore at our disposal for the enhancement of the proposed constitutive model. However, the present study aims at describing the cyclic behaviour for high-cycle fatigue life predictions, with as few parameters as possible. We thus consider that such improvements of the model are not crucial in our case. At low or moderate stress levels, it appears indeed that the unloading curves and the ratchetting are correctly predicted: the error between the experiment and the simulation becomes substantial only above 100 MPa, which represents a high stress level in a fatigue context. At last, a better knowledge of the microstructural mechanisms is very helpful for a relevant construction of the internal variables involved in a phenomenological approach. Moreover, it should also influence the choice of a fatigue criterion designed for short glass fibre reinforced thermoplastics. 5.3. Discussion about the hysteretic areas

Computed dissipated energy per cycle (mJ/mm3)

R The hysteretic loops, which stand for the dissipated energy per cycle W diss ¼ cycle rde, are rather well predicted for the CCR test, the TRR tests and the strain-controlled tensile–tensile test (see Fig. 27). For the stress-controlled tensile–tensile tests (2.5 and 250 MPa/s), hysteretic areas are slightly under-estimated by the proposed model. As suggested by Klimkeit (2009) and Klimkeit et al. (2010), an energetic criterion might be relevant for predicting the fatigue duration of SGFR thermoplastics. This criterion, inspired from the pioneering work of Morrow (1965), has been formerly used in order to evaluate the low cycle fatigue life of metallic materials (Charkaluk and Constantinescu, 2000; Skelton, 1991). Klimkeit et al. demonstrate the purpose of an energetic approach for reinforced thermoplastics, but are not able to split the mechanical energy into elastic or plastic (dissipated) parts. The need of a nonlinear behaviour constitutive law is reckoned for the determination of the plastic strain energy. Our model appears to be well predictive regarding this physical quantity, which might be an asset for the future research of a fatigue criterion relying on the local mechanical state. 1

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It is also possible to take advantage from the thermal measurements in order to identify the dissipative mechanisms (Chrysochoos et al., 1989; Luong, 1998). Such a work requires the development of a fully coupled thermo-mechanical constitutive model in the GSM framework. Taking into account this supplementary information into the subsequent model might lead to a better estimation of the dissipated energy. 6. Conclusions In this paper, we have investigated the mechanical behaviour of a SGFR polyamide 66 under complex tensile loadings. We have displayed an extended experimental campaign achieved on conditioned material (RH50) at room temperature. Thanks to a large range of mechanical histories and loading rates, in accordance with service life conditions, we have highlighted several mechanical features which are taken into account in the modelling of the cyclic behaviour. The proposed constitutive equations therefore describe:  a short-term viscoelasticity, underlined by the stress-rate sensitivity of the initial stiffness;  a long-term viscoelasticity, which occurs at low stress levels with creep, stress relaxation or strain recovery without any remanent strain mechanism;  a non-linear viscous flow rule, combined with a non-linear kinematic hardening law, which models the residual strain evolution according to the relaxed stress, as well as the viscous (called pseudo-viscoplastic) effects at moderate or high stress levels;  a cyclic softening which is activated by the dissipated pseudo-viscoplastic energy. The model parameters have been determined by means of an identification methodology which uncouples at best the different mechanical features. The fair agreement between numerical simulations and experimental data shown on the tests which are not involved in the identification process proves that the proposed model is relevant for predicting the cyclic behaviour of the studied material under complex tensile loadings. The same experimental campaign conducted on DAM material has been accurately simulated thanks to the same constitutive equations, with another set of optimized parameters. It means that the proposed model can be successfully applied below (DAM material) and around (RH50 material) the glass transition temperature. In both cases, it remains nevertheless difficult to predict the unloading paths at high stress levels. Such a problem has already been mentioned in the literature for unreinforced polymeric materials (Ayoub et al., 2010; Colak, 2005; Drozdov and Christiansen, 2003; Rémond, 2005) and its solving would certainly require a distinction between the loading or unloading cases in the nonlinear viscous flow rule or in the yield function. Such a point needs further work but has not been considered as a major shortcoming for fatigue applications. The first outlook of this work lies in the investigation of the mechanical behaviour for extended humidity rate and temperature conditions. It would be particularly interesting to estimate if the proposed constitutive model would be still predictive above the glass transition temperature, which regards many applications for the automotive industry. This work is in progress and the first results are rather encouraging. The influence of the fibre orientation represents of course a major point in order to study industrial components for which the orientation distributions are complex. The anisotropy of the overall mechanical behaviour depends on the local orientation distribution and its study will require to test samples with different fibre orientations. This work will include the dependence of material properties on the microstructure characterized by its second-order orientation tensors aw, which reads Ce ¼ Ce ðaw Þ, P ¼ Pðaw Þ. Note that the numerical simulation of an industrial structure requires the knowledge of the fibre orientation distribution field, which implies that the material properties are heterogeneous. This will be discussed in a forthcoming paper. The final aim of this study is the prediction of fatigue life under cyclic loadings. It will consist in suggesting a fatigue criterion based on the knowledge of the local mechanical history. Taking advantage of the accurate description of the complex mechanical features above-mentioned, such a criterion will benefit from the prediction of non-linear physical quantities, such as the dissipated energy, the cumulative viscoplastic strain or the cyclic softening. In the same manner, this prediction ability enables us to analyze complex loadings including interruptions or creep–fatigue coupling. Acknowledgements This work was supported by PSA Peugeot-Citroën and has received the financial support of the French Minister for Research (ANRT). The authors gratefully acknowledge DuPont de Nemours for providing material data and specimen. DuPont™, and all products denoted with Ò or™ are registred trademarks of E. I. du Pont de Nemours and Company or its affiliates. References Advani, S., Tucker III, C., 1987. The use of tensors to describe and predict fiber orientation in short fiber composites. Journal of Rheology 31, 751. Andriyana, A., Billon, N., Silva, L., 2010. 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