1

PREDICTION OF LONG TERM BEHAVIOUR OF COMPOSITEMATERIALS

RUI MIRANDA GUEDESDEMEGI, Faculdade de Engenharia Mecânica e Gestão IndustrialRua dos Bragas, 4099 Porto CodexJOSÉ J. L. MORAISUTAD, Departamento de EngenhariasQuinta dos Prados, 5000 Vila RealANTÓNIO TORRES MARQUESDEMEGI, Faculdade de Engenharia Mecânica e Gestão IndustrialRua dos Bragas, 4099 Porto CodexALBERT H. CARDONDepartment of Mechanics of Materials and Construction (MEMC), VrijeUniversiteit Brussel (VUB), Pleinlaan 2, B-1050 Brussels, Belgium

Abstract

After a brief introduction, where the question of durability related to compositematerials is raised concerning the type of applications and the constituents of thecomposite material, some considerations are given about the interaction betweenphysical and chemical ageing and its effect on the long term behaviour of thosematerials.

A comparison of different methodologies to define models for the constitutive lawsof the material is made, together with a reference to methodologies used for realstructures in order to predict long term behaviour.

An algorithm to predict long term laminate properties of fibre reinforcedcomposites is presented. In the procedure, the classical laminate plate theory wasextended to include time-related response of composite materials for membrane andbending load. An efficient semi-direct time integration scheme, providing a stableintegration process, was derived to be included in the numerical procedure. Thepresent formulation enables solving problems related with creep, stress relaxation andrate dependent stress-strain behaviour for in-plane and bending loads.

The results concerning studies related to the influence of water and temperature onthe long-term behaviour of fibre reinforced composite, hence on the constitutivemodels, are presented. The experimental results shows that a combined time-temperature-moisture superposition principle (TTMSP) is not valid. This principlesimply states that the effects of temperature and moisture on the viscoelastic propertiescan be uncoupled.

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1. Introduction

Fibre reinforced composite materials have been used for quite a long time in highlydemanding applications, where static and/or dynamic loads are applied in the presenceof environmental loading (e.g. temperature, water or other aggressive liquid). Until notlong ago the main structural applications could be seen in aeronautical and aerospatialsituations with very specific type of environments. More recently, the use of compositematerials was extended for primary structures in aircrafts, automotive applications,infrastructures, rebars and rehabilitation/strengthening of bridges and even completebridges. This facts brings the issue of durability, hence prediction of long termproperties and residual life, as a determinant factor in the success of the referredapplications. There is a strong need to define methodologies for the constitutive modelsto predict long term behaviour of composite materials, and for the short term test ofcomposite materials structures which may “validate” the prediction i.e. may be a short-term safeguard for the structure.

In composite materials one may identify the resin and the interface fibre/resin asthe elements with more influence in time dependent properties. Besides resin and thefibre/resin interface may act as a protection to the fibres as far as environmental effectsare concerned. However, when organic fibres are used (e.g. aramidic fibres,UHMWPE), the variation of fibre properties with time is also a factor to be consideredin the prediction of long term properties of composite materials.

The long-term behaviour of composite materials may be affected by physical (e.g.changing in Tg) and chemical ageing (change in molecular weight, oxidation, changein density of reticulation). In fact, the swelling, the plasticization and the slowhydrolysis (with scission) of the resin, and the slow attack of the liquid to thefibre/resin interface correspond to loss of properties with influence in the creep andfatigue behaviour of composite materials.

Although some research has been done Struik [1], Pasricha [2], Zhang [3],Milkovich [4], Peng [5], much has to be done in order to integrate the environmentaleffects into the constitutive equations of viscoelasticity.

2. Comparison of methodologies

2.1. CONSTITUTIVE MODELS

Dillard [6] and Brinson [7] defined a methodology for the acceleratedcharacterisation of composite materials towards the prediction of residual life at a longterm. This can be summarised in figure 1.

3

Short term testsfor the plypropertiescharacterization

Creepcompliances

Sij (time,Temp.,

σ ,...)

Time toRupture

σ tr ( σ ,Temp.,...)

long termconstitutive lawsof the laminate

Long termrupture modelof the laminate

Rupturecriteria

Durability prediction for generallaminated composites

Long termbehaviour modelfor the ply

Long termpredictions

SchaperyEquation

CLT or FEM

Figure 1. Methodology to characterise the long-term behaviour of general laminatecomposite materials using short term tests.

Other methodologies, with different scientific basis, have been presented. They canbe summarised as follows:a) Rupture analysis as function of time based in energetic terms, Hiel [8].b) The critical element concept, Reifsnider [9] - identifies an element of volumerepresentative as “critical” in the sense of catastrophic failure and all the phenomenaand models used for the determination of residual strength will come as a continuousfunction of time.c) Talreja [10] characterises indirectly damage (i.e. microdamage) associating a set ofinternal variables representative of the state (damage) of the material.d) Ladvèze [11] also characterises indirectly damage, by its association with the effectin the global modulus of the region with the distributed damage entities, i.e. thedeterioration indicator is related with the relative variation of elastic stiffness.e) Cardon [12] suggests that a combination of methodologies proposed by Ladvèze orTalreja with the Time-Temperature-Stress Superposition Principle as proposed byBrinson [7] based on Schapery theory, would give better predictions.

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2.2. SHORT-TERM TEST OF STRUCTURES

In several applications, such as pipelines, the expected life is, at least, 50 years. Themethodologies for prediction of long-term behaviour are based on tests ofrepresentative samples together with statistical extrapolations. Several methodologiesexist, such as dynamical loading method, ultimate elastic wall stress method, strain atfailure method Brito [13] but all of them imply tests with duration up to 10 000 hours.

Short-term tests have been recently proposed [13] and [14] where a short-term testmade with the support of acoustic emission, will enable to define a strain level belowwhich there is a guarantee that the long term restrictions imposed are assured.

3. A Model for the long term predictions

In order to predict the long-term behaviour of the basic material component, a researchprogram based on the idea proposed by Hal Brinson [15] was developed. The basis ofthe method of accelerating factors in viscoelastic and viscoplastic models was the workdone by Schapery [16]. The accelerating factors considered were the stress and thetemperature. In future more factors should be included such as the moisture, the ageingeffects and the mechanical degradation. The prediction of the long-term durability ofcomposites must be based on the coupled effects of all these non-linear mechanisms. Abrief discussion of that research program is given bellow.The single integral formulations, such as the Schapery non-linear viscoelastic theory,thought simple in form, have been found to be mathematically too complex to be usedin an engineering analysis, due to the presence of Volterra-type integrals. Tuttle andBrinson [17] presented a closed-form solution to an arbitrary number of discrete stepsin stress based upon the Schapery theory. This solution, based on hereditary nature,requires that all stresses at each time step are stored and reused for all subsequent timesteps. Furthermore, each additional time step requires the hereditary integral to berecalculated from the initial starting time. Therefore, the total solution time growsgeometrically with the number of time steps that ultimately limits the number of timesteps, regardless the size of the step or the speed of the computer. An efficient methodto handle the constitutive integral equation of Schapery theory is presented. Themethod avoids storing the stress history and computes the Volterra-type integral byreplacing it with numerical equivalent ordinary integrals. The proposed method, tohandle the constitutive integral equations of Schapery theory, can be easily used topredict the stress relaxation and the rate-dependent stress/strain behaviour.

3.1. ANALYTICAL MODEL DESCRIPTION

The predictions for multiple-angle laminates can be obtained using the classicallaminate plate theory. The constitutive models are defined at the ply level, which is a"macromechanic" approach. The plane of a single ply is defined by the 1-2 coordinatesystem, where the 1 and 2 axes are parallel and perpendicular to the fibres,

5

respectively. Consider a ply subjected to a plane stress state { }122211 ,, τσσ that may

change with the time. The total strains induced within the ply are given by

( )( )( )

( )( )

( )( )( )

( )( )

VPt

t

t

t

t

tS

tSS

SS

t

t

t

+

=

12

22

12

22

11

66

2221

1211

12

22

11 0

00

0

0

γε

τσσ

γεε

,

(1)

where the subscript VP denotes the viscoplastic component of the strain. We assumed

that the fibre-dominated compliance terms 11S and ( )2112 SS = are time-independent,

while the matrix dominated compliance terms ( )tS22 and ( )tS66 are time-dependent.

The total strains associated with the matrix-dominated compliances 22S and 66Swere modelled using the modified Schapery theory to include the viscoplasticbehaviour, firstly presented by Tuttle [18]. According to this, the total ply strainsinduced by an arbitrary stress history are given by

( )( )( )

( )( )( )

( ) ( )( )

( ) ( )( )( )

( )

+

−∆

′−∆

+

=

∫∫

∫

∫66

66

2222

0 1266

0 2222

0

1266,26666,1

0

2222,22222,1

12

22

1

66,066,0

22,022,012

1211

12

22

11

0

´

0

00

0

0

nt N

nt N

t

t

dC

dC

dd

gdSg

dd

gdSg

t

t

t

Sg

SgS

SS

t

t

t

τττ

ττσ

ττ

ττϕϕ

ττ

τσψψ

τσσ

γεε

,

(2)

where 222222 ,, nNC and 666666 ,, nNC are stress-independent but temperature

dependent material viscoplastic properties. The kernels ( )tS22∆ and ( )tS66∆ are the

transverse and shear elastic compliance, respectively, with the correspondent reducedtimes ´,´,, ϕϕψψ given by

6

∫∫′

=′

=τ

σσ

τψτψ

0 22,

0 22,

´ , a

d

a

dt

,

(3)and

∫∫′

=′

=τ

σσ

τϕτϕ

0 22,

0 22,

´ , a

d

a

dt

,

(4)

where 22,22,222,122,0 ,,, σaggg and 66,66,266,166,0 ,,, σaggg are stress-dependent

non-linear parameters. The common parameter for the transverse and shear non-linearcompliance is the octahedral shear stress in the matrix that is a function of matrixtransverse stress and matrix shear stress. A more detailed explanation of the octahedralshear stress parameter is given by Schapery [16].In order to eliminate the Volterra-type integrals, the transverse and shear complianceare expressed using Prony series, as Gramoll [19] and Czyz [20] have alreadydiscussed.

( ) ( )( ) ( )

−=∆

−=∆

∑

∑

=

−

=

−

n

i

ti

n

i

ti

i

i

eStS

eStS

1

66,66

1

22,22

66,

22,

1

1

λ

λ

,

(5)

where 66,66,22,22, ,,, iiii SS λλ are linear viscoelastic parameters for the transverse and

shear compliance, respectively. For example substituting (5) into (2) for the transversestrain, we obtain

( ) ( ) ( ) ( )ttgtSgt vp

n

ii 22,

122,22,12222,022,022 εεσε ++= ∑

= ,

(6)where

( ) ( )( )∫ ′−−−=t

ii dd

dgeSt i

0

2222,222,22,

22,1 ττσ

ε ψψλ.

(7)

7

After some transformations, Equation (7) gives the relation between the internal

strain ( )122, +ji tε and ( )ji t22,ε . Let us assume a linear variation during 1j+j ttt ≤≤for the stress, 22σ , as

( ) ( )t

tt

dt

d jj

∆−

= + 2212222σσσ

.

(8)

Let us also assume that the function ( )σ22,~

ig is also approximated by a linear

variation as

( ) ( )( ) ( )( )t

tgtg

dt

gd jijii

∆−

≅ + 2222,12222,22,~~~ σσσ

,

(9)where

( ) 2222,222,22, ~ σσ gSg ii = .

(10)

Substituting Equation (10) into Equation (7) we obtain

( ) ( )( ) ( ) ( ) ( )∫ −−−= j

ijit it

jiji dd

gdeetgt

0

22,2222,22,

~~ 22,22, τ

τσ

σε τθλθλ ,

(11)

where ( ) ∫′

=t

a

dt

0 22,σ

τθ .

In order to calculate ( )122, +ji tε in terms of ( )ji t22,ε we obtain

( ) ( )( ) ( ) ( )[ ] ( )( ) ( )[ ]( ) ( )[ ] ( )

∫+ +

+

−−

−−++

−

−−=

1 122,

122,

22,

22,2222,12222,122,

~

~~

j

j

ji

jji

t

t

it

jiji

tt

jiji

dd

gde

ttgetgt

ττ

σεσσε

τθθλ

θθλ

.

(12)

8

The integral of Equation (12) can be solved using the approximation function (9) as

( ) ( )[ ] ( )

( )( ) ( )( )

( )( )

( )( )

−×

×

∆−

=

=

+

+ +

∆−

+

+

−−∫

12222,22,

1 122,

1

~~

~

22,

12222,

2222,12222,

22,

ji

j

j

ji

ta

t

i

j

jiji

t

t

it

e

ta

t

tgtg

dd

gde

σλ

σ

τθθλ

σ

λσ

σσ

ττ

σ

,

(13)

where jj ttt −=∆ +1 and the following approximation was used

( ) ( )[ ] ( )

( )( )( )τσλ

στλτθθλ

σ

τσ

−≅

′−=−−

++

+ ∫+

112222,

22,

2222,

22,122,

1

jj

i

t

iji

tta

a

dt

j

.

(14)

Substituting Equation (10) into Equation (9) we obtain

( ) ( ) ( ) ( )( )

( ) ( )( )jitt

i

jit

iji

tji

tgeet

tget

tet

ii

ii

22

12222,122,

~11

+

~11

1

ση

ση

εε

ηη

ηη

−−

∆

−

∆−+⋅=

∆−∆−

+∆−∆−

+

,

(15)where

( )( )12222,

i,22

a +

=j

i tσλ

ησ

.

(16)

9

Tuttle et al. [18] presented the solution of the viscoplastic term assuming a linearvariation in stress over every time step as

( ) ( )[ ] ( )[ ]

( ) ( )[ ] ( )[ ]( ) ( ) ( )[ ]( )

222222

221

2222

12212222

122

1122

22j

2222

0 2222122

1t=

n

jjjj

Nj

Nj

nt

t

Nt N

VPj

ttttN

ttC

dCdCtj

j

j

−

−⋅+−

+

+=

++

+++

+ ∫∫+

σσσσ

β

ψψσψψσε

,

(17)with

( ) ( ) ( )[ ] ( )[ ]( ) ( ) ( )[ ] ( )1

1222222

1122

122

221-jj 1tt

2222

−−

+−

+

−−+

−+= jj

jj

Nj

Nj

ttttN

ttC

σσσσ

ββ .

(18)

The total strain for each layer, given by Equation (2), can then be rewritten as

( ){ } [ ] ( ){ } ( ){ }111 +++ += jjelastj tRtSt σε ,

(19)

where

( ){ } ( ) ( )

( ) ( )

+

+=

+=

+

+=

++

∑

∑

16,1

16,66,1

12,1

12,22,11

0

jvp

n

iji

jvp

n

ijij

ttg

ttgtR

εε

εε .

(20)

The vector of Equation (20) contains the viscoelastic and viscoplastic deformationsas given by Equation (2). This vector is calculated recursively using the formulas givenby Equation. (15) and Equation (17) since they can equally be applied into transverseand shear deformations.

The objective is to determine the behaviour of multidirectional composites knowingthe stress-strain relationships for unidirectional plies. Therefore a modified laminate

plate theory for laminates subjected to in-plane loads { }N and out-plane-loads { }Mwas developed. The subscript ( )zyx ,, represents the off-axis coordinate directions

(laminate or global coordinates).

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Assuming that the ply strain is linear in the thickness coordinate z, the total strainof the ply k in the global coordinate system is given by

( ){ }( )( )( )

( )( )( )

+

=

+

+

+

+

+

+

+

1

1

1

10

10

10

1

jxy

jy

jx

k

jxy

jy

jx

kj

tk

tk

tk

z

t

t

t

t

γεε

ε ,

(21)

where { }0ε is the in-plane laminate strain vector, { }k is the laminate curvature

vector and kz represents the z coordinate of the kth ply. Taking into account the

in-plane loads and moments acting on the laminate, the equilibrium equations inglobal coordinates are

{ } { }{ } { }

=

=

∫∫−

−

Mdzz

Ndzh

h

h

h2

2

2

2

σ

σ ,

(22)

where h represents the total laminate thickness. If we assume a constant stress at eachply, which is not true for flexural loads, using Equations. (19), (21) and (22) we obtaina system of equations that allow determining the in-plane laminate strain, the laminatecurvature and the stress state in each ply. In order to avoid large systems of equations,the ply stress state given by Equation (19) can be used in Equation (22) to obtain thefollowing condensed system of equations as

[ ] [ ]

[ ] [ ]{ }{ }

{ }{ }

[ ] { }

[ ] { }

+

=

+

∑

∑

∑∑

∑∑

=

−

=

−

=

−

=

−

=

−

=

−

p

kkkkkelast

p

kkkkelast

p

k

kkkkelast

p

kkkkelast

p

kkkkelast

p

kkkelast

hzRS

zRS

M

N

khhzShzS

hzShS

1

1

1

1

0

1

321

1

1

1

1

1

1

12

ε

,

(23)

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where kh represents the thickness of kth ply. The previous formulation enables

solving the creep problem. If a restriction is imposed, like prescribing an in-planestrain or a curvature, then the relaxation problem must be solved. In a creep test theloads are imposed and the in-plane strain and curvature are determined. In a relaxationtest thein-plane strain and/or curvature are imposed and the resulting loads are determined.The existence of a valid solution is briefly discussed. In strict mathematical terms, eachrestriction implies an inversion of the correspondent non-linear integral equation andnumerical methods are the only way to obtain the solution. Hence this formulationenables solving problems related with creep, relaxation and rate dependent stress/strainbehaviour for in-plane and flexural loads. The method described here was developed ina FORTRAN computer program called LAMFLU.

3.2. MATERIAL PROPERTIES CALCULATION

A numerical procedure to fit the Schapery parameters was developed. Lou andSchapery [21] described a mixed procedure, graphical and numerical, to determine allthe parameters of Schapery model. In this case a full numerical approach wasdeveloped in order to avoid the inherent subjectivity of the graphical analysis. A plasticmodified Schapery model was chosen as it was proposed and applied by Tuttle andPasricha [22].

The material properties were obtained through a series of creep and creep-recoverytests. The stress history in a creep and creep-recovery consists of a constant stress

( 0σ ) applied during the creep phase followed by removing the total applied stress

during the creep-recovery phase of the test. During creep time ( att ≤≤0 ) the strain

response is given by

( ) ( )∑=

−

+

−+=

m

i

nNa

t

icreep tCeDgggDti

100

21000 1 σσσελ

σ .

(24)

During the recovery time ( att > ) the response is given by

( ) ( ) ( )nNm

i

a

t

ttieryre tCeeDgt

ia

iaa00

1

2cov 1 σσελ

λ σ +

−= ∑

=

−−−

.

(25)

Data collected from creep and creep-recovery tests at several stress levels in

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combination with a fully numerical procedure were used to obtain the materialsproperties.

In this method the problem was made linear by prescribing the { } mii ,,2,1 �=λparameters. The first step is the estimation of the irrecoverable viscoplastic straininduced during creep at various stress levels. The recovery curves given by Equation(25) can be rewritten in the following compact form

( ) ( )∑=

−−+=m

i

ttieryre

iaeAAt1

0covλε ,

(26)

where iAA ,0 are parameters to be determine for each stress level. The recovery data

is numerically fitted to Equation (26) using one time constant per decade of time in thelogarithmic scale. From the previous formulation it is possible to extrapolate therecovery curves for very long times, calculating the limit

( ) ( )naN

eryret

tAt 00cov C lim σε ==∞→

.

(27)

The product nN ⋅ is determined by curve fitting of Equation (27). From the lowerstress level, assumed to be a linear level, the values of the linear viscoelasticcompliance are determined using Equations (25) and (26) as

( )iati

ie

AD λσ −−

=10

.

(28)

The higher stress levels given by jσ are then used to determined the parameters

2g and σa as

01

2 =

−−

− ia

a

t

iji eDgAλ

σσ .

(29)

13

The values for 0D , C and n are evaluated by fitting the linear recovery data to

Equation (25). Finally the parameters 0g and 1g are evaluated by fitting the non-

linear creep data to Equation (24).

The present method transforms a non-linear problem into a linear problem byprescribing the parameters that make the problem non-linear, in this case the

{ } mii ,,2,1 �=λ parameters. The remaining parameters are determined by linear systems

of equations obtained by the Least Square Method. This procedure avoids the ill-conditioning problem but the solution depends on the prescribed parameters.

The linear viscoelastic creep compliance and the viscoplastic flow are the onlytime-dependent material properties in the present model. The other material propertiesdepend on the stress and on the temperature. The material properties were expressed asfunctions of the octahedral shear stress in the matrix. The method described wasdeveloped into a FORTRAN computer program called XAPFIT.

3.3. EXPERIMENTAL RESULTS

Some experimental results are presented for tests conducted on composite materialsunder creep and creep-recovery, relaxation and ramp loading.

In Figure 2 shear strain history for the test data and the LAMFLU prediction areplotted for a T300/5208 [±45°] laminate. The cyclic creep and creep-recovery tests atroom temperature were conducted for a total time of 10176 hours, representingapproximately 14 months. A very good agreement between the prediction andmeasured data over the entire test was obtained for the theoretical model.

0

5000

10000

15000

20000

25000

30000

0 2000 4000 6000

τxy = 46.85 MPa

she

ar s

tra

in,µε

T300/5208

8000 10000time, hours

12000

test data

LAMFLU

Figure 2. Test data and LAMFLU prediction shear strain of a T300/5208 [±45°]

14

laminate subjected to a cyclic creep and creep-recovery test at room temperature.

The LAMFLU procedure was used to compute the relaxation shear stress fordifferent constant shear strains extending them into the non-linear range. The resultingrelaxation behaviour predicted from the creep data showed a good agreement with theactual shear stress relaxation behaviour as shown in Figure 3.

0

10

20

30

40

50

0.01 0.1 1 10 100

T300/5208

she

ar

stre

ss, M

Pa

test dataLAMFLU9000 µε

time, hours

8000 µε4000µε

1500 µε600µε

Figure 3. Shear stress relaxation of T300/5208 [±45°] laminates under constantshear strain at room temperature, and predicted with LAMFLU.

In Figure 4, test data and respective plastic shear strain are plotted for a shearstrain rate of 1250µε/s together with the LAMFLU prediction. Comparison betweenthe experimental result and the curve based on the theoretical model show an excellentagreement.

15

0

20

40

60

80

100

0 0.001 0.002 0.003 0.004 0.005 0.006

test data

LAMFLU

time, hours

T300/52081250µε/s

shea

r st

ress

, M

Pa

Figure 4. Constant shear strain rate test of T300/5208 [±45°] laminate at roomtemperature, with those predicted with LAMFLU.

In Figure 5 the maximum strain history is plotted for four specimens in a four pointbending test together with theoretical prediction. The test program consists on 6 cycles,3 steps to increase the load followed by another 3 steps to decrease the load. Goodagreement with experimental data was obtained for the first three load-steps. For theload-decreasing phase, the LAMFLU predictions showed a fair agreement withexperimental data with deviations lower than 15%.

0.0000

0.0010

0.0020

0.0030

0.0040

0 200 400 600 800 1000 1200 1400time, hours

Max

imum

str

ain,

ε

specimen 1

specimen 2

specimen 3

specimen 4

LAMFLU

Figure 5. Maximum strain results and theoretical predictions for four point bendingtest.

16

A numerical method to handle the constitutive integral equation of the Schaperytheory was developed. The procedure can be used to predict the creep for in-plane andbending load, stress relaxation and rate-dependent stress/strain behaviour of compositelaminates. A good agreement between computed and experimental data was observedin the presented examples. The plastic modified Schapery model proposed by Tuttle[18] appear to be a general model at least for the limited stress or strain states andhistories used in this study.

The Schapery equation and the associated procedure to determine the equationparameters are an open problem, i.e., there is not a unique solution. Nevertheless theXAPFIT program always gives the same solution if the imposed parameters

{ } mii ,,2,1 �=λ are the same. One can also argue that the solution obtained from creep

and creep-recovery may not be valid for another load history such as stress relaxation,multiple steps loading, for example. However the present results and the resultspresented by Tuttle [22] have denied this argument.

5. Environmental effects

The viscoelastic analysis presented above includes only the mechanical loading. Inthe actual service conditions the mechanical loads act in combination with theenvironmental factors, like moisture, temperature and radiation. This coupled actiongives rise to a complex phenomenon at the molecular level inside the material, whichinduces changes in its macroscopic viscoelastic properties. The understanding of thosephenomena and the modelling of the time property evolution is a crucial task for long-term durability analysis of a polymer-based composite materials. Before trying toovercome the complexity of the synergistic effects of mechanical loading, moisture,temperature and radiation, it is necessary to analyse the isolated influence of thesefactors.

A great concern exists about the moisture effects on polymer-based composites,which can be grouped in two main categories: chemical and physical effects. The post-cure reactions, hydrolysis and leaching of low molecular weight species are some of thechemical effects reported in the literature Antoon and Koenig [23], Piggot [24]Broutman [25], Vauthier et al. [26], Bouten and deWith [27]. The physical effects ofmoisture are the well known plasticization and swelling of polymer matrix Browninget al.. [28], McKagu [29], De Iasi and Whiteside [30], Hahn and Kim [31]. Thesephenomena are coupled and in certain circumstances are the origin of microstructuraldamage like fibre debonding and matrix cracking Weitsman [32].

For a given material system and hygrothermal condition it is important to know thecontributions of chemical, physical and damage mechanisms, induced by moisture, tothe current mechanical response of a composite. However, it is not clear whatexperimental methodology one should adopt to achieve this goal and, consequently, themodelling of moisture effects on the mechanical behaviour remains an open problem.Even when the hygrothermal conditions are such that only the physical mechanisms

17

are active, the moisture influence is very difficult to model. The results of anexperimental investigation performed by Morais [33] will be presented and discussedbellow, in order to illustrate some problems to be resolved in the analytical modellingof the physical effects of hyrothermal ageing.

The material system used in the experimental work was a quasi-unidirectionalglass/epoxy composite supplied by PPG in a prepreg form, under the trade name

PV 245®. The material was exposed at 75°C and several hygroscopic conditions (0%,24%, 34%, 73% and 92% relative humidity). Two exposure times were selected:1600 hours (the time to achieve moisture equilibrium) and 3100 hours. Under thisageing times the chemical degradation mechanisms are inactive, as was detected byfast Fourier transform-infrared spectroscopy. However, the reduction of the glasstransition temperature (Tg) after exposure (Figure 6), measured by dynamicthermomechanical analysis (DMTA), is an indicator of the physical mechanismsoperating at the molecular scale due to moisture absorption. The physical effects areonly partially reversible, indicating a reorganisation of the internal material structureafter an absorption/desorption cycle. The existent models for the Tg dependencerelative to the moisture concentration, such as the Fox rule of mixtures Nielsen andLandel [34], cannot predict this behaviour:

gm

m

gr

rgw T

w

T

wT += .

(32)

In this equation Tgw , Tgr and Tgm are the glass transition temperatures of the wetmaterial, dry resin and water, respectively, and wr and wm are the weight fractions ofresin matrix and absorbed moisture, respectively.

1801601401201008060402000

400

800

1200

1600

2000

1600 hours3100 hoursAbs./desor.Dry

Temperature (°C)Figure 6. Influence of ageing time on Tg.

18

The quasi-static tensile tests of a unidirectional laminate in the transverse direction(figure 7 and table 1) gives additional information about the influence of thehygrothermal ageing in the internal microstructure change of the material. In fact, thestress-strain response is completely reversible after an absorption/desorption cycle,suggesting the absence of microstructural damage. The microscopic examination of thespecimen edges (before testing) and the fracture surfaces also support this conclusion.So, the combined interpretation of quasi-static tensile tests and the DMTA tests (figure6) points for a rearrangement of the molecular network of the matrix resin, resulting ina greater free volume.

0.0070.0060.0050.0040.0030.0020.0010.0000

10

20

30

40

50

60

70

80

Dry1600 hr1600 hr./ desorption

3100 hr/desorption3100 hr.

StrainFigure 7. Hygrothermal ageing effects on the stress-strain curve in transverse

direction.

Another salient feature displayed by the quasi-static mechanical response is itsdependence on the exposure time. As can be seen in figure 7 and table 1, an importantchange in the stress-strain curve occurs after 3100 hours of hygrothermal conditioning.This time dependent response cannot be related to the variation of Tg, which remainsconstant between 1600 and 3100 hours of exposure (figure 6). Therefore, the empiricalequation of Chamis et al. [35] is inappropriate to model the effect of moisture on themechanical properties of a composite, even when are only in activity the physicalmechanisms. That equation has the form:

2/1

00

−−

=TT

TT

P

P

gw

gw,

(33)

19

where P is a wet composite property at temperature T and P0 is its value correspondingto the reference temperature T0. The initial (before ageing) stress-strain curve ispractically recovered after desorption (figure 7). In view of this, a possible explanationfor the relation between mechanical behaviour and ageing time is the viscoelasticchange of the stress and strain micro-fields, promoted by matrix plasticization,Morais [33]. The slight discrepancies between the original stress-strain curve and therecovered one after 3100 hours of ageing can be attributed to the hysteresis ofviscoelastic behaviour.

TABLE 1. Hygrothermal effects on the transverse Young modulus (σ2) and transversestrength (X2).

Materialcondition Initial 1600 hr

1600 hr

desorption3100 hr

3100 hr

desorption

E2 (GPa) 19.52 18.49 19.55 15.87 18.79

σ2 (MPa) 103.0 98.3 101.6 62.0 96.4

To deal with the influence of temperature and moisture on the viscoelastic responseof polymeric matrix composites it was proposed a combined time-temperature-moisturesuperposition principle (TTMSP). It was assumed that the effects of temperature andmoisture can be uncoupled and that a time-temperature-moisture shift factor aTH can bedefined as Daniel et al. [36], McCullough [37]:

( ) ( )caTaa HTTH ⋅= ,

(34)

where aT is a time-temperature shift factor (depending only on the temperature T) andaH is the time-moisture shift factor (depending only on the moisture concentration c).To verify the validity of this assumption the master curves of storage modulus based ontime-temperature superposition principle (TTSP) and time-moisture superpositionprinciple (TMSP) are constructed (figure 8), for distinct reference temperatures and thesame moisture concentration were worked out. The shift factor aT is ineffective to bringthe two curves into coincidence, thus invalidating the equation (34).

20

1050-5-10-15-20-253.4

3.6

3.8

4.0

4.2

4.4

original/50°C0%/40°C0%/50°C

logΦ (Hz)

Figure 8. Master curves from the TTSP, TMSP and TTMSP.

An additional element of complexity in the moisture effects modelling is illustratedin figure 9, were the master curves of storage modulus in the original dry state andafter an absorption/desorption cycle are compared. The irreversible behaviour observedis related to the changes of aT(T) with hygrothermal ageing. This is probably theunderlying cause, which renders the equation (34) inadequate.

1050-5-10-15-20-253.2

3.4

3.6

3.8

4.0

4.2

4.4

dry73%

logΦ (Hz)

Figure 9. Master curves of the storage modulus in the initial dry state and after anabsorption/desorption cycle.

21

6. Conclusions and future research

The Schapey’s model modified to include a viscoplastic element together with thealgorithm to adjust experimental parameters gave good results. The algorithmovercomes the ill-conditioned problem that arises from form the adjustment ofexponentials. Hence a standard procedure for the experimental data adjustment can beprepared based upon this algorithm. In future all laboratories should adopt a standardapproach. The neuro-fuzzy approach developed by Y. Qin [38] to characterise thenon-linear viscoelastic and viscoplastic behaviour of composite materials seems to be agood basis for the standard approach. This new approach treats all test stress levelstogether and the non-linear parameters are obtained directly in a close form, thereforethe overall accuracy is improved.

A numerical procedure, LAMFLU, was developed to solve problems of laminatedplates using the classical laminate plate theory and the Schapery constitutive integralequation. The procedure enables solving time-dependent related problems forlaminated plates with in-plane and bending loads. The versatility of LAMFLUalgorithm and the ability to predict long-term behaviour of composite structures underdifferent types of loading has been shown, although some discrepancies were detectedin the recovery phase of complex loading.

Environmental effects altered significantly the material’s behaviour with theviscoelastic change of the stress and strain micro-fields, due to matrix plasticization.These effects are not reversible after an absorption/desorption cycle, probably due to aninternal reorganisation of the material structure. The influence of moisture andtemperature are not equivalent, neither additive, i.e., the combined time-temperature-moisture superposition principle (TTMSP) is not valid.

Based upon the acquired experience a future research programme is proposed.Develop an algorithm to associate the damage theories of Talreja and Ladevèze withthe Schapery constitutive model for non-linear viscoelastic materials, allowing a betterunderstanding of viscoplasticity and its relation to damage and better accuracy inprediction. Develop a method for long-term prediction of physical and chemicalageing. Based on chemical composition and environmental interaction, identify type ofmatrices which possibly behave according to the proposed models, and introduce theenvironmental changes in the constitutive laws for the relevant materials.

22

7. References

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3. Zhang, Li, (1995). Time-Dependent Behaviour of Polymers and UnidirectionalPolymeric Composites . PhD thesis, Delft University of Technology.

4. Milkovich, S.M., Herakovich, C.T., and Sykes, G.F., (1986). Space RadiationEffects on the Thermo-Mechanical Behaviour of Graphite-Epoxy Composites.Journal of Composite Materials, Vol. 20, November.

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12. Cardon, A. H., Helmelrijck, D. Van, Hiel, C. C., Qin, Y., (1993). Long TermBehaviour of Polymer Based Composite Systems Under Complex Loading.Proceedings of the International Conference on Advanced Composite Materials,Edited by T. Chandra and A. K. Dhingra, The Minerals, Metals and MaterialsSociety.

13. Brito, F. J. M. M., Marques, A.T. (1997) Comparison of methodologies forprediction of long-term properties of GRP pipes. Proceedings of the InternationalConference Progress and Analysis of Composite Systems, Edited by K.Reifsneider, D. Dillard and A. Cardon, Balkema, pp. 289-294.

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15. Brinson, H.F. and C. Yeow 1978 A New Experimental Method for theAcceleratedCharacterisation of Composite Materials, Proceedings of the 6th Int. Conf. OnExperimental Stress Analysis, Munich.

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