Modelling the influence of temperature and relative

humidity on the time-dependent mechanical behaviour

of a short glass fibre reinforced polyamide

Antoine Launay, Yann Marco, Habibou Maitournam, Ida Raoult

To cite this version:

Antoine Launay, Yann Marco, Habibou Maitournam, Ida Raoult. Modelling the influence oftemperature and relative humidity on the time-dependent mechanical behaviour of a shortglass fibre reinforced polyamide. Mechanics of Materials, Elsevier, 2013, 56, pp 1 - 10,issn = ”0167-6636”, url = ”http://www.sciencedirect.com/science/article/pii/S016766361200.<10.1016/j.mechmat.2012.08.008>. <hal-00778275>

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https://hal-polytechnique.archives-ouvertes.fr/hal-00778275

Submitted on 19 Jan 2013

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Modelling the influence of temperature and relative humidity on the time-dependentmechanical behaviour of a short glass fibre reinforced polyamide

A. Launaya,b,c,∗, Y. Marcoa, M.H. Maitournamb, I. Raoultc

aLaboratoire Brestois de Mecanique et des Systemes (EA 4325 ENSTA Bretagne/UBO/ENIB), 2 rue Francois Verny, 29806 Brest Cedex 9, FrancebLaboratoire de Mecanique des Solides (CNRS UMR 7649),Ecole polytechnique, 91128 Palaiseau, France

cPSA Peugeot Citroen (Direction Scientifique et des Technologies Futures), Route de Gisy, 78943 Velizy-VillacoublayCedex, France

Abstract

Polymer matrix composites, and especially short glass fibrereinforced polyamides, are widely used in the automotive industry.Their application on structural components requires a confident mechanical design taking into account the sensivity oftheir me-chanical response to both temperatureT and relative humidityH. In this paper, the constitutive model already developed bytheauthors (Launay et al., 2011) is successfully applied to describe the non-linear time-dependent behaviour of a PA66-GF35 undervarious hygrothermal conditions. The extensive experimental database involves testing conditions under and above the glass tran-sition temperatureTg. An equivalence principle between temperature and relative humidity is proposed and validated, since thenon-linear mechanical response is shown to depend only on the temperature gapT − Tg(H).

Keywords: PMCs, polyamide, non-linear constitutive behaviour, temperature, relative humidity

1. Introduction

Carmakers are facing the challenge of reducing CO2 emis-sions, and are thus strongly interested in light materials as analternative to metallic materials. Polymer matrix composites(PMCs) and especially short glass fibre reinforced (SGFR) ther-moplastics are a cost-efficient solution which combines a suf-ficient stiffness for many automotive components subjectedtomechanical loadings, as well as a large freedom of shapes pro-vided by the injection moulding. Some of these componentsare located in the air-intake circuit (e.g. intake manifolds, in-let gas compressor exit), for which the choice of a polyamidematrix offers a sufficient thermal strength. These parts undergocyclic loadings, induced by mechanical sources (intern pulsedpressure), in a complex hygrothermal environment. The tem-peratureT varies according to the engine torque-regime range;the relative humidityH is the direct consequence of climaticvariations. Both temperature and relative humidity are knownto have an impact on mechanical properties of the polyamidematrix (Valentin et al., 1987; Guerin, 1994; Verdu, 2000).

Fatigue design of automotive components generally requirestwo steps. The first one regards the determination of the cyclicbehaviour: constitutive equations are needed to describe themechanical response of the material for any loading signal rep-resentative of service conditions. The model must be rich enoughto capture (as fairly as possible) the physical features whichmay have an influence on the fatigue life (stress/strain ampli-tudes, cumulated strain, dissipated energy, etc). This is areasonwhy a phenomenological model has been proposed for SGFR

∗Corresponding author. Tel.: +33 1 57 59 41 66; fax: +33 1 41 24 32 56Email address:antoine.launay@mpsa.com (A. Launay)

thermoplastics in a former paper (Launay et al., 2011). Thismodel has been successfully validated on cyclic loadings andcomplex loading histories with stress levels up to 80% of UTS.The second step relies on the knowledge of the local mechanicalstate: a physical quantity is shown to be correlated to the fatiguelife under various loading conditions (Launay et al., 2012b,a).As the fatigue strength of SGFR polyamides is highly sensitiveto both temperature and relative humidity (Bernasconi et al.,2007; Sonsino and Moosbrugger, 2008), it is essential to in-vestigate the influence of hygrothermal conditions on theirme-chanical response. This is the topic of the present paper, forwhich we focus on a polyamide 66 reinforced with 35 wt% ofshort glass fibres (PA66-GF35).

Some authors have investigated the hygrothermal ageingof thermoplastic composites, and more specifically of SGFRpolyamide (Bergeret et al., 2001; Thomason et al., 2010). How-ever, we do not consider degradation mechanisms in this study;we rather focus on the activation of deformation mechanismsinduced by the temperature or the humidity rate.

The thermal sensitivity of thermoplastic materials has beenwidely discussed in the literature; many works are based onthe review of deformation mechanisms of semi-crystalline ther-moplastics published byBowden and Young(1974) and latersupplemented byLin and Argon(1994). Crystal plasticity inlamellae stacks is thermally activated since the temperature low-ers the critical resolved shear stress, whereas chain reptationsin the amorphous phase are “frozen” below the glass transi-tion temperature (the amorphous phase is in its glassy state) andhighly activated above (rubber-like state). This remarkable tem-perature, denotedTg, plays an important role on the mechani-cal properties of polymeric materials: the Young modulus de-

Preprint submitted to Elsevier June 8, 2012

creases dramatically between the glassy and rubber-like stateswhereas the ultimate strain increases, viscous dissipative phe-nomena are mostly prominent aroundTg, etc. This is the rea-son why some authors developed constitutive models for neatpolymers valid either below or aboveTg (Mulliken and Boyce,2006; Boyce et al., 2000). Only few models (e.g.Dupaix and Boyce,2007, for amorphous polymers) are able to capture the complexbehaviour of polymers over a wide range of temperature includ-ing Tg. On the other hand, one should notice recent develop-ments of the cooperative model introduced inFotheringham and Cherry(1978)’s works on linear polyethylene.Richeton et al.(2005)indeed take advantage of a strain rate-temperature superpositionprinciple to predict the yield stress of amorphous polymersfora large temperature range (under and aboveTg) and for quasi-static to impact loading rates.Gueguen et al.(2008) extendedthe model to semi-crystalline polymers. However, such modelscannot describe loading-unloading paths, and are thus of lessinterest regarding fatigue applications.

The influence of the relative humidity on the mechanical re-sponse is a phenomenon specific to polyamides, because of thehydrophilic feature of the amide functional groups (Puffr andSebenda,1967). A partial pressure of water in the air1 results in sorp-tion of water molecules by the polyamide matrix. The diffu-sion of these molecules in the polymeric network has a nega-tive influence on its mechanical cohesion since it lowers theH-bonds strength and thus makes the chain mobility easier. Thisis reported in the literature as the “plasticizing” effect of water(Merdas et al., 2002): the ultimate strain increases, the mate-rial stiffness decreases. Water molecules are mainly absorbedby the amorphous phase (Guerin, 1994), which explains whyanother consequence of relative humidity is the decrease oftheglass transition temperature of hygroscopic polymers likepolyamides(Kettle, 1977).

Different equations have been proposed in the literature tomodel the evolution ofTg according to the equilibrium watercontentw, or according to the relative humidityH – see thework of Smith and Schmitz(1988) for a short review of differ-ent models, and also (Reimschuessel, 1978; Ellis et al., 1983).However, these equations are not very satisfying for the caseof the polyamide 66 (see Fig.1). They are either not accu-rate, like the Fox equation – its non-accuracy has also beenobserved on PA6 byKettle (1977) – and the Kelley-Buecheequation, or require physical parameters which are not easilydetermined (this is the case of the Reimschuessel and the Ellis-Karasz equations). Note that these four equations also requirethe determination of a preliminary relationship betweenw andH (because the relative humidity is the physical quantity whichis controlled), which reads:

w = a Hb (1)

wherea and b are two material parameters. This equation isvalid for a wide range of hygrometric conditions, i.e.H ∈ [0, 90]%(Verdu, 2000). At least four parameters are thus needed for the

1The relative humidity is defined as the ratio between the partial pressure ofwater and the vapor pressure:H = 100 ×

pH20

pS.

−20

0

20

40

60

80

0 20 40 60 80 100

Gla

ss tr

ansi

tion

tem

pera

ture

(°C

)

Relative humidity (%)

Exp. dataLinear interp.Fox eq.Ellis−Karasz eq.Kelly−Bueche eq.Reimschussel eq.

Figure 1: Comparison between different equations modellingthe influence of the relative humidity on the glass transitiontemperature. Experimental data (courtesy of DuPont™) are de-termined by DMA tests.

Reimschuessel or the Ellis-Karasz equations. From an indus-trial point of view, it is more convenient to work withH, whichis easily measured or monitored, rather than withw, which re-sults from complex sorption-diffusion mechanisms. When plot-tingTg againstH (see Fig.1), a linear equation appears to fairlydescribe the decrease of the glass transition temperature accord-ing to the equilibrium relative humidity:

Tg(H) [○C] = T 0

g − χ ⋅H [%] (2)

T 0

g is the glass transition temperature of dry PA66, andχ is theslope. Albeit being completely empirical, this very simplefor-mulation (involving only two parameters) gives the best results,which is the reason why it will be used in the following of thispaper.

The outline of this paper is as follows. Section2 presentsthe material of this study as well as the experimental protocols.In Section3, the constitutive model (seeLaunay et al., 2011)is applied to a large range of hygrothermal conditions whichcorrespond to the complex environment encountered during theservice life. In Section4, we propose an equivalence principlebetween the temperature and the relative humidity, the influenceof both parameters being condensed through the difference be-tween the temperatureT and the glass transition temperatureTg(H). This principle and its consequences regarding the evo-lution of material parameters are discussed in Section5, andsome conclusions and outlooks are exposed in Section6.

2. Experimental

2.1. Material and samples

The material of this study is a polyamide 66 reinforced with35 wt% of short glass fibres (PA66-GF35), and provided byDuPont™ under the commercial name Zytel® 70G35 HSLX

2

Polyamide 66 (23○C, DAM)Molecular weight2 34.0 kg.mol−1

Crystallinity rate2 35.6%Melting temperature2 263○CYoung modulus 3063 MPaPoisson coef. 0.36

Glass fibresVolumic fraction 19.5%Average fibre length 250×10−6 mAverage fibre diameter 10×10−6 mYoung modulus 72000 MPaPoisson coefficient 0.22

Table 1: Physical and mechanical properties of the studiedPA66GF35.

R60

112

10 20

80

4

170

Figure 2: Injection moulded ISO527-2-1A tension specimen(all dimensions are in mm).

(see Tab.1 for a few mechanical characteristics). This mate-rial is currently used for automotive application, especially forcomponents of the air intake circuit. ISO527-2-1A tensile spec-imens (see Fig.2) were injection moulded from this grade. Lo-cal strain has been measured with a biaxial contact extensome-ter (model EPSILON™ 3560-BIA-025M-010-HT1 fitted for atemperature range from -40 to 150○C). Tensile load has beenapplied by a INSTRON™ 1342 hydraulic machine equippedwith mechanical grips, and controlled by a 100 kN capacityload cell.

Different environmental conditions (T -H) have been tested.For each of them, the tensile samples have been conditionedat the equilibrium with an air containingH% of relative hu-midity. They have been weighted to ensure that water equi-librium is reached. Tests at room temperature have been per-formed in standard conditions, the test durations being shortenough to neglect any variation of the water content. On theother hand, tests at temperatureT from 50 to 140○C have beenperformed in a climatic chamber, where both temperature andrelative humidity are monitored. It is indeed commonly admit-ted that diffusion mechanisms of water molecules in the poly-meric network are thermally activated, obeying an Arrheniuslaw (Verdu, 2000; Merdas et al., 2002). The samples have alsobeen weighted before and after each mechanical test to ensurethat the water content remains constant during a test and is con-sistent with the conditioning.

PA66-GF35 (23○C)H (%) 0 20 27 33 50 72 75w (%) 0.00 0.40 0.65 0.89 1.74 3.14 3.35

Table 2: Equilibrium water contentw as a function of relativehumidityH for the studied material. No significant sensitivityto the temperature has been observed.

2.2. Tensile tests

Figure3 sums up the different mechanical tests that havebeen conducted for each environmental condition. An exten-sive experimental database has been build up, since eleven dif-ferent conditions have been investigated, representativeof dif-ferent service life conditions on vehicles (see also Fig.6):

• three conditions regard dry-as-molded (DAM) material,at temperatureT = 23, 110, 140○C;

• six conditions are performed at room temperature (23○C),with DAM material (already mentioned) and with mate-rial conditioned at the equilibrium with relative humidityH = 20, 27, 33, 50 and 75%;

• three conditions require the use of the climatic chamber:T=60○C andH=50%,T=90○C andH=50%,T=70○CandH=72%.

Table2 specifies the equilibrium water content for each humid-ity condition. Presented data are in accordance with Eq.1,under the assumption that glass fibres do not absorb moisture.

2.3. Dynamic mechanical analysis (DMA)

Dynamic mechanical analyses on ISO527-2-1A specimenshave been conducted for different water contents on the IN-STRON™ 1342 hydraulic tensile machine. Temperature sweepsat frequencyf = 1 Hz have been performed in climatic cham-ber, the relative humidity being controlled in order to ensure aconstant water content in the samples. DMA is usually con-ducted on samples with a small cross-section, which is not thecase of ISO527-2-1A specimens. This is the reason why ther-mal stabilization steps (during 30 minutes) have been respectedfor each temperature level so that the thermal gradient throughthe specimen might be neglected. The microstructure for in-jected reinforced plastics is strongly coupled to the geometry,this is why it is crucial to perform DMA tests on the same sam-ples as static (and fatigue) tests.

A tensile load signal of small amplitude is applied on thespecimen. The glass transition temperature is usually defined asthe temperature corresponding either to the maximum value ofthe loss factortan δ, or to the inflexion point of the evolution ofthe storage modulusE′. According to the literature, these twomethods generally give similar results, which seems consistentwith our tests (see Fig.4).

2These data have been provided by DuPont™. The crystallinityrate and themelting temperature are determined by DSC (Differential Scanning Calorime-try).

3

σ(t)

t

Monotonic tensile tests,stress controlled

σ(t)

t

Cyclic creep-recovery (CCR)test, stress controlled

ε(t)

Anhysteretic curve test,strain controlled

stressrelaxation

creep

strainrecovery

t

Figure 3: Definition of mechanical tests of the experimentalstudy. All tests have been carried out with a tensile load, either stressor strain controlled.

−20 0 20 40 600.6

0.8

1

1.2

1.4x 10

4

Temperature (°C)T

S

tora

ge

modulu

s (M

Pa)

E’

−20 0 20 40 600.02

0.03

0.04

0.05

0.06

−20 0 20 40 600.02

0.03

0.04

0.05

0.06

L

oss

fac

tor

tan

δ (−

)

Figure 4: Result of DMA performed on conditioned material(H=75%).

Figure5 shows the influence of the relative humidity (wa-ter content of the sample being at the equilibrium) on the glasstransition temperature. As expected,Tg decreases whenH in-creases. Note that the value of the glass transition tempera-ture depends on the loading conditions: experimental data fromDuPont™ performed on the same material (but in flexural modeon another sample geometry) indicates thatTg ∼ 80

○C for DAMmaterial, whereasTg ∼ 30

○C for material conditioned atH=50%.The slope is thus similar, but the absolute value is higher, whichmay be explained by a sensitivity of viscoelastic effects onthelocal fibre orientation distribution. In the present work, thesame samples are used for DMA and tensile tests, in order toavoid any additional complexity due to the fibre orientationdis-tribution.

3. Application of the proposed constitutive behaviour

3.1. Identification of material parameters

Constitutive equations for the cyclic behaviour of SGFRthermoplastics have been proposed and validated for the presentmaterial in a previous paper (Launay et al., 2011), for two hy-grothermal environments: DAM material (H=0% andT=23○C)and standard conditions (H=50% andT=23○C). As mentionedpreviously, nine other hygrothermal conditions have been stud-ied, leading to a rich database. Note that these various envi-ronments are chosen under, above or around the glass transitiontemperature, in order to cover at best the different cases encoun-tered during the service life on vehicules (see Fig.6).

The equations are not re-written here. The one-dimensionalrheological scheme is just pictured in Fig.7 and we recall thatthe mechanical strainε is split in four parts:

4

−30−20−10

0 10 20 30 40 50 60

0 20 40 60 80 100

Gla

ss tr

ansi

tion

tem

pera

ture

(°C

)

Relative humidity (%)

Exp. dataLinear interp.

Figure 5: Evolution of the glass transition temperature of thestudied PA66-GF35 in respect with the relative humidity, andlinear interpolation.

0

20

40

60

80

100

120

140

0 10 20 30 40 50 60 70 80

Tem

pera

ture

(°C

)

Relative humidity (%)

Glass transition

Tests at 23 °CTests on DAM material

Complex conditions

Figure 6: Sum up of all experimental conditions investigated inthe present study. The dotted line illustrates which conditionsare above or under the glass transition.

• a viscoplastic strainεvp

, which obeys a non-linear kine-

matic hardening and also a non-linear thresholdless flowrule, similar toDelobelle et al.(1996)’s model;

• a long-term viscoelastic strainεv1

(with characteristictime τ1);

• a short-term viscoelastic strainεv2

(with characteristictime τ2 ≪ τ1);

• an elastic strainεe, associated to softenable elastic mod-

uli (the softening function being activated by the energydensity dissipated through viscoplastic mechanisms).

The identification strategy is the same as the one describedin (Launay et al., 2011). The initial Young modulusE0

e as wellas the short term viscoelastic parameters (Ev2, η2) are char-acterized thanks to the sensitivity of the initial stiffness to thestress-rate (at the beginning of monotonic tensions). Long-termviscoelastic parameters (Ev1, η1) are then identified on creep-recovery histories at low loading levels, i.e. for which no ir-recoverable strain has been observed (at the beginning of theCCR test). Viscoplastic parameters (A, H, m, C, γ) are iden-tified on the loading stage of the anhysteretic curve. At last, thesoftening parameters (a, b) stem from a compromise betweenthe results obtained on the anhysteretic curve and the mono-tonic tests, which may also require a slight correction on vis-coplastic parameters. One sees that only five mechanical tests(three monotonic tensions at different stress rates, the CCR testand the anhysteretic curve) are needed for the determination ofthe 12 material parameters ; this process is unambiguous thanksto the uncoupling between the different strain components.Thesecond part of the CCR test (from the activation of irrecover-able mechanisms until failure) stands for a validation testof theoverall identification strategy.

3.2. Validation and simulation of the mechanical response

Figure8 displays the comparison between the experimen-tal data and the results of the proposed model for monotonictensions in various environmental conditions. The agreementis very good over the whole range of hygrothermic conditions,even if only seven different ones are plotted for the sake of clar-ity. The transition from a quite stiff response (DAM materialat room temperature) to a softer response with high strains tofracture (high temperature or high relative humidity) is espe-cially very well described. The proposed model is thus relevantfor the simulation of the mechanical response of componentslocated in the air-intake circuit, where the hygrothermal envi-ronment is complex.

Figures9 and10 confirm that the complexity of the non-linear time-dependent behaviour is well captured by the pro-posed model, under or above the glass transition temperature.Creep, recovery or relaxation steps are well simulated, even forstress levels reaching 70% of UTS. Remark that the end of theCCR test (from the end of the first cycle att=1100 s until fail-ure att=5500 s) is a full validation case, which highlights theaccurate prediction of the proposed model.

5

A,H,m

C, γ

η1

Ev1 Ev2

η2

E0e , a, b

εvp εv1 εv2 εel

σσ

Figure 7: 1-D rheological scheme of the proposed model for the cyclic behaviour of SGFR thermoplastics (Launay et al., 2011).

0

50

100

150

200

250

0 1 2 3 4 5

Str

ess

(MP

a)

Strain (%)

T23 RH0T23 RH20T23 RH33T23 RH50T23 RH75T60 RH50T140 RH0

Figure 8: Application of the constitutive model to mono-tonic tensile tests (25 MPa/s) for different hygrothermal con-ditions. Experimental (symbols) and numerical (continuouslines) curves are ordered as indicated in the legend.

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0 1000 2000 3000 4000 5000 6000

Str

ain

(%)

Time (s)

T23 RH33

CCR testModel

Figure 9: Application of the constitutive model to the cycliccreep recovery (CCR) test at 23○C on material conditioned atH=33%.

−20

0

20

40

60

80

100

0 2000 4000 6000 8000 10000

Str

ess

(MP

a)

Time (s)

T140 DAMAnhysteretic curveModel

Figure 10: Application of the constitutive model to the anhys-teretic curve at 140○C on DAM material.

It is thus possible to simulate fatigue loadings for variousenvironmental conditions, even if high loading levels and com-plex hygrothermalconditions imply that non-linear mechanismsare activated. This result is very interesting regarding the ap-plication of a fatigue criterion (Launay et al., 2012a), whichrequires the accurate description of the local mechanical re-sponse.

4. Temperature-humidity equivalence

The influence of both temperature and relative humidity onthe time-dependent behaviour meansa priori that an exten-sive experimental database is required for confident fatigue de-sign. However, as aforementioned, both have similar effectson the mechanical response, which is the consequence of therole played by the amorphous phase during the deformation ofthe semi-crystalline thermoplastic matrix. FollowingMiri et al.(2009) who investigated the tensile behaviour of neat polyamide6 in different hygrothermal conditions, it seems that the me-chanical response is mainly controlled by the distance to theglass transition temperature, called temperature gap,T−Tg(H).

The present section is thus dedicated to the experimentalinvestigation of an equivalence principle between temperature

6

and relative humidity regarding the mechanical response ofthematerial of the study.

4.1. Equivalence for viscoelastic properties

It is possible to illustrate the temperature-humidity equiva-lence on (short-term) viscoelastic properties through theresultsof DMA tests. Figure11 indeed shows a good superpositionof the evolution of the storage modulusE′ as a function of thedistance to the glass transition temperatureTg, for each relativehumidityH. The evolutions of the loss factortan δ accord-ing toT − Tg reveal a less statisfying agreement (see Fig.12).This may be explained by the experimental measurement of thephase angle between load and displacement signals, which ismore complicated because of the low amplitude of the imposedload signal compared to the 100 kN capacity of the load cell.

At low stress levels, the constitutive model may be reducedto a classical first-order Kelvin-Voigt model. It enables the ana-lytical expression ofE′ andtan δ as functions of the short-termviscoelastic (Ev2, η2) and elastic (E0

e ) parameters for any sinu-soidal stress signal of pulsationω:

E′ =(E0

e +Ev2)E0

eEv2 + (η2ω)2E0

e

(E0e +Ev2)2 + (η2ω)2

(3)

tan δ =η2ω(E

0

e)2

(E0e +Ev2)E0

eEv2 + (η2ω)2E0e

(4)

Figure11displays a good agreement between viscoelastic char-acteristics from the constitutive model (the parameters ofwhichbeing identified as presented in Section3) and the experimentalcurves, especially regarding the storage modulusE′. The dif-ferences observed on the loss factor (see Fig.12) are due to theintrinsic difficulty of representing viscoelastic properties withthe same set of material parameters at both low (DMA tests)and higher (monotonic tensions) stress levels.Detrez(2008)faced the same problem on unreinforced thermoplastic mate-rials: the viscoelastic properties characterized on DMA testscannot be directly used in his constitutive model for the cyclicbehaviour of semi-crystalline thermoplastics.

In spite of this remark, these first results indicate the goodapplication of an equivalence principle on viscoelastic proper-ties.

4.2. Equivalence for non-linear properties

Among all hygrothermal conditions we investigated duringour experimental campaign, two are similar regarding the dis-tance to the glass transition temperature. The samples testedatT=90○C andH=50% are supposed to present a temperaturegap of +74○C, compared toT − Tg=+72○C for those tested atT=70○C andH=72%. The gapT − Tg(H) is computed withthe empirical equation2 characterized in Section2 (see Fig.5).

Figures13 and14 show that the non-linear mechanical re-sponse in both hygrothermal conditions are indeed very similar.The monotonic tensile curves (Fig.13) at different stress ratesare well superposed over a wide range of stress levels. This re-sult is very comparable to those presented byMiri et al. (2009)

6000

8000

10000

12000

14000

−40 −20 0 20 40 60 80

Sto

rage

mod

ulus

E’ (

MP

a)Temperature gap (°C)

Model param.Exp. : RH10Exp. : RH20Exp. : RH50Exp. : RH65Exp. : RH75

Figure 11: Evolution of the storage modulusE′ according tothe parameterT − Tg(H).

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

−40 −20 0 20 40 60 80

Loss

fact

or ta

n δ (

−)

Temperature gap (°C)

Model param.Exp. : RH10Exp. : RH20Exp. : RH50Exp. : RH65Exp. : RH75

Figure 12: Evolution of the loss factortan δ according to theparameterT − Tg(H).

7

0

20

40

60

80

100

120

0 1 2 3 4 5

Str

ess

(MP

a)

Strain (%)

Stress rate: 2.5 MPa/s

25 MPa/s

190 MPa/s

Monotonic tension, 70°C RH72Monotonic tension, 90°C RH50

Figure 13: Experimental illustration of the temperature-humidity equivalence principle on monotonic tensions(T − Tg ∼ 73

○C).

on neat PA6. The results on CCR tests (Fig.14), which illus-trate all the complexity of the mechanical behaviour (creep, re-covery, accumulated plastic strain, cyclic softening) are, to theauthors’ knowledge, completely original in comparison to theliterature. They are very interesting since they let us affirm thatthe good agreement between the monotonic tensions in bothhygrothermal conditions is not the mere result of a compen-sation between different deformation mechanisms (e.g. plas-tic and viscous ones). The complexity of the loading historyshows that the strain evolution during creep or recovery stepsis the same, and that the irrecoverable strain accumulationandthe stiffness decrease are activated in a very similar way inbothcases. The temperature-humidity equivalence principle isthere-fore validated regarding the long-term viscoelastic properties aswell as the non-linear viscoplastic properties.

5. Discussion

5.1. Influence of hygrothermal environment on material param-eters

The evolutions of the different parameters of the proposedconstitutive model are plotted in Fig.15 as functions of thetemperature gap. In spite of a small scattering, the trends arequite regular, which confirms that the influence of both tem-peratureT and relative humidityH may be described throughonly one parameter,T − Tg(H). The glass transition results inimportant variations of model parameters, which is consistentwith the important role played by the amorphous phase duringdeformation.

The analysis of the parameters evolutions shows that at lowT−Tg values, the mechanical response is nearly elasto-viscoplastic.On the other hand, in the neighbourhood of the glass transi-tion (T − Tg ∼ 0), viscoelastic features are very important andthe cyclic softening is activated. WhenT − Tg is greater, the

mechanical behaviour is essentially viscoplastic, since the ap-parent viscoelastic surface is reduced (H is very small); more-over, the decrease of the hardening moduli occurs simultane-ously with higher material ductility.

The temperature-humidity equivalence principle thereforehelps us to understand the evolutions of the material parametersover a wide range of hygrothermal conditions. It appears thatthe proposed constitutive model is useful to describe all featuresof the mechanical response for applications on the air-intakecircuit, but it could also be simplified for restrained rangeofhygrothermal conditions.

5.2. A worthwhile equivalence principle

From the industrial point of view, such an equivalence prin-ciple is very useful in order to make easier the mechanical char-acterization of polyamide materials. The experimental databasemay be reduced to the investigation of the relative humidityin-fluenceor of the temperature influence, provided that the rela-tionshipTg = Tg(H) has been determined. The latter point isnot time-consuming since a linear relationship has been shownto be sufficient for the studied material.

5.3. Crystalline phase

As already mentioned byMiri et al. (2009), the equivalencebetween the effects of temperature and relative humidity onde-formation mechanisms is well understood regarding the amor-phous phase. From the macroscopic point of view, temperatureand moisture indeed play the same role since they both makeeasier the chain mobility in the amorphous network (Verdu,2000; Merdas et al., 2002). On the other hand, temperature alsoactivates plastic mechanisms in the crystalline phase of semi-crystalline thermoplastics (Bowden and Young, 1974; Lin and Argon,1994), whereas water molecules are not supposed to diffuse inthe crystalline lamellaes. Two hypotheses may be formulated toexplain that the equivalence principle seems yet to be relevant:

• water molecules may penetrate in stack faults in the crys-talline phase and thus affect its mechanical properties(Miri et al., 2009);

• the presence of the short glass fibres limits the strain lev-els: the deformation mechanisms are mainly concentratedin the amorphous phase and the crystalline lamellae areassimilated to elastic inclusions, their role during non-linear deformation is thus secondary.

This point requires further work and experimental investiga-tions at the local scale in order to explain the surprising factthat the crystalline phase does not seem to prevent the validityof the temperature-humidity equivalence. Since the studied ma-terial is a short fibre reinforced polyamide, the influence ofbothtemperature and moisture on fibre-matrix interface should alsobe studied: non-linear deformation mechanisms may as well beexplained by fibre debonding or cavitation at fibre tips.

8

0

20

40

60

80

100

120

0 1 2 3 4 5

Str

ess

(MP

a)

Strain (%)

CCR test, 70°C RH72CCR test, 90°C RH50

(a) Stress versus strain response

0

1

2

3

4

5

0 1000 2000 3000 4000 5000 6000 7000 8000

Str

ain

(%)

Time (s)

CCR test, 70°C RH72CCR test, 90°C RH50

(b) Strain versus time response

Figure 14: Experimental illustration of the temperature-humidity equivalence principle on cyclic creep recovery (CCR) tests(T − Tg ∼ 73

○C).

6. Conclusion

In this paper, we have studied the influence of the hygrother-mal environment on the mechanical behaviour of a SGFR polyamide(PA66-GF35) subjected to complex loading-unloading histo-ries. On the basis of a recently proposed model (Launay et al.,2011), it was shown that the approach enables the descriptionof the mechanical response over a wide range of hygrothermalconditions. A temperature-humidity equivalence principle hasbeen underlined since the gap between the temperature and theglass transition temperature appears to control the sensitivityof the time-dependent behaviour to both temperature and rela-tive humidity. This principle has been experimentally assessedwith DMA, monotonic and complex loading-unloading tensiletests. It is valid for temperature ranging from 23 to 140○C andrelative humidity ranging from 0 to 75%, that is to say for tem-perature gapsT − Tg(H) from -40 to +80○C. Such a result hasa strong interest regarding the non-linear mechanical character-ization of polyamide matrix composites for a broad range ofhygrothermal conditions, which is required for reliable fatiguedesign.

Acknowledgements

This work was supported by PSA Peugeot-Citroen and hasreceived the financial support of the French Minister for Re-search (ANRT). The authors gratefully acknowledge DuPontde Nemours for providing material data and specimen.

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0.0e0 5.0e5 1.0e6−40

−20

0 20

40 60

80

5.0e3 1.0e4 1.5e4

Viscoelastic moduli (MPa)

Elastic modulus (MPa)

Tem

perature gap (°C)

Ev1 (left)

Ev2 (left)

Ee (right)

1.0e5 1.0e6 1.0e7 1.0e8−40

−20

0 20

40 60

80

1.0e3 1.0e4 1.0e5 1.0e6 1.0e7

Long−term viscoelastic parameter (MPa.s)

Short−term viscoelastic parameter (MPa.s)

Tem

perature gap (°C)

η1 (left)

η2 (right)

1.0e−11 1.0e−9 1.0e−7 1.0e−5−40

−20

0 20

40 60

80

0 10 20 30 40 50

Charact. strain rate (1/s)

Charact. stress (MPa)

Tem

perature gap (°C)

A (left)

H (right)

0.0 2.0 4.0 6.0 8.0−40

−20

0 20

40 60

80

Viscoplastic exponent (−)

Tem

perature gap (°C)

m (left)

0.0e0 2.0e4 4.0e4 6.0e4 8.0e4 1.0e5−40

−20

0 20

40 60

80

0.0e0 2.0e2 4.0e2 6.0e2 8.0e2 1.0e3

Hardening modulus (MPa)

Nonlinear parameter (−)

Tem

perature gap (°C)

C (left)

γ (right)

0.0 5.0 10.0 15.0 20.0 25.0 30.0−40

−20

0 20

40 60

80

0.0 0.1 0.1 0.2 0.2 0.3 0.3

Maximal softening (%)

Charact. energy density (mJ/mm3)

Tem

perature gap (°C)

a (left)b (right)

Fig

ure

15

:V

ariation

of

the

mo

delp

arameters

accord

ing

toth

etem

peratu

reg

apT−Tg (H).

Th

esym

bo

lsstan

dfo

rth

eid

entified

param

eters,and

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ind

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ds.

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