a modelling of the coupled thermodiffuso-elastic linear

Oil & Gas Science and Technology – Rev. IFP, Vol. 58 (2003), No. 5, pp. 571-591Copyright © 2003, Éditions Technip

A Modelling of the CoupledThermodiffuso-Elastic Linear Behaviour.Application to Explosive Decompression

of PolymersG. Rambert1, J.C. Grandidier1, L. Cangémi2 and Y. Meimon2

1 Laboratoire de mécanique et physique des matériaux, ENSMA, 1, avenue Ader, BP 40109, 86961 Futuroscope Cedex - France2 Institut français du pétrole, 1 et 4, avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex - France

e-mail: [email protected] - [email protected] - [email protected] - [email protected]

Résumé— Modélisation du comportement thermodiffuso-élastique linéaire couplé. Application à ladécompression explosive de polymères— Une modélisation thermodiffuso-mécanique est proposée icidans le but de décrire le comportement couplé de polymères initialement placés dans un environnementgazeux et sujets à une décompression « explosive ». Plusieurs descriptions du volume élémentairereprésentatif (VER) pouvant répondre à cet objectif sont détaillées. La modélisation est ensuite conduitesur la représentation la plus simple du VER comme un mélange homogène de polymère et de gaz. Dansle cadre de la thermodynamique des milieux standard généralisés, les lois constitutives couplées sontétablies en se limitant à un comportement mécanique élastique linéaire. L’implantation de ce modèledans le logiciel ABAQUS™ a alors permis de mener une première étude qualitative des effets decouplages directs dont les principaux résultats sont discutés.

Abstract— A Modelling of the Coupled Thermodiffuso-Elastic Linear Behaviour. Application toExplosive Decompression of Polymers— A thermodiffuso-mechanical modelling is proposed in thisarticle with the aim of describing the coupled behaviour of polymers, which are initially placed in agaseous environment and subjected to an “explosive” decompression. Some descriptions of theelementary representative volume (ERV) are considered and detailed. The modelling is then developedfrom the simplest representation, which consists in describing the ERV as a homogenous mixture ofpolymer and gas. In the framework of generalized standard media, coupled constitutive equations areonly proposed for a linear elastic behaviour. The numerical implementation of this model in ABAQUS™led to a first qualitative study of the direct couplings. Its main results are discussed here.

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Oil & Gas Science and Technology – Rev. IFP, Vol. 58 (2003), No. 5

NOTATIONS

General and Mathematical Notations

★ quantity associated to the mixture★ 0 initial quantity associated to the mixture★ g (★ p) quantity associated to gas (polymer)

★ 0g (★ 0

g) initial quantity associated to gas (polymer)

★→

vector★= second order tensorI= identity tensor

time derivative

partial derivative at ★ constant

grad gradient operatordiv divergence operatortr trace operator★→

· ★→

scalar product

Thermodynamic Notations

ψ specific free energy (J·kg–1)s specific entropy (J·K–1·kg–1)s0 initial specific entropy (J·K–1·kg–1)

J→s entropy flux (W·K–1·m–2)

Φ volume dissipation (W·m–3)Φ1 volume dissipation linked to temperature gradient

(W·m–3)Φ2 volume dissipation linked to chemical potentials

gradient (W·m–3)d dissipation potential (W·m–3)d*, d*

i dual dissipation potentials (W·m–3)

Mechanical Notations

ur radial displacement (m)γ→ acceleration of the considered particle in the

mixture (m·s–2)f→

body force per unit of mass (N·kg–1)σ= total Cauchy stress tensor (Pa)σ=0 initial total Cauchy stress tensor (Pa)ε=

e infinitesimal elastic strain tensorλ, µ Lamé coefficients (Pa)E Young coefficient (Pa)ν Poisson coefficient

Thermal Notations

T absolute temperature (K)T0 initial absolute temperature (K)

J→

q heat flux per unit area (W·m–2)r volume density of heat generated by an external

source (W·m–3)Λ thermal conductivity (W·m–1·K–1)Cε=

e,ci

specific heat at constant strain and normalizedconcentrations (J·K–1·kg–1)

Diffusion Notations

ρ average mixture density (kg·m–3)ci normalized concentration of the i th constituent

(Pa)c0

i initial normalized concentration of the ith cons-tituent (Pa)

µi mass chemical potential of the i th constituent(J·kg–1)

µ0i initial mass chemical potential of the ith constituent

(J·kg–1)J→

mirelative mass flux of the ith constituent (kg·m–2·s–1)

si solubility coefficient of the ith constituent in themixture (Pa–1)

D diffusivity (m2·s–1)

Coupling Notations

αT isotropic heat expansion coefficient (K–1)αc coefficient of expansion linked to diffusion

phenomenonkµ coefficient linked to the chemical potential

gradient effect on the gas mass flux (kg·s·m–3)kT coefficient accounting for the thermal gradient

effect on the entropy flux (W·m–1·K–2)cTµ coefficient corresponding to the temperature

(chemical potential) gradient effect on the mass(entropy) flux (kg·m–1·s–1·K–1)

d coefficient representing the effect of temperaturevariation (concentration variation) on the chemicalpotential (entropy) (J·kg–1·K–1)

D coefficient defined as (ρD/kµ) (m2/s2).

INTRODUCTION

During the past twenty years, the use of polymers expandedwidely in technical structures. In petroleum industry forinstance, they form sheaths and gaskets used for oiltransportation. They are exposed to severe in-serviceenvironment, defined by high temperature (80 to 150°C) andby a gas pressure, which fluctuates abruptly between high(100 MPa) and low values. These drops in pressure (calledexplosive decompression) sometimes induce irreversibledamage in the polymer as a result of a full coupling betweenmechanical, thermal and diffusion phenomena. Depending

∂∂

★

d

dt

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G Rambert et al./ A Modelling of the Coupled Thermodiffuso-Elastic linear Behaviour

on the nature of the gas-polymer system, decompression rateand thermal conditions, such damage appears either locally(mesocracking) or in a diffuse way (foaming). Predictivetools, taking these couplings into account, should be used toforesee the in-service behaviour of such structures. The workin progress is based on the development of constitutive laws.The latter would make it possible to report the differentinteractions between mechanical, thermal and diffusionphenomena.

From an experimental viewpoint, the explosive decom-pression phenomenon was often studied, particularly in thecase of elastomers, which are often employed in the oilindustry and are not very resistant to explosive de-compression. A few examples are given hereafter forinformation: the works of Briscoe et al. (1992, 1993, 1994),Campion (1990) and Liatsis (1989). General investigationswere also made on thermoplastics by Dewimille et al.(1993)and Jarrin et al. (1994), more particularly on polyethylene ina methane environment (Gaillard-Devaux, 1995) and poly-vinylidene fluoride with carbon dioxide (Lorge et al., 1999).All these studies confirm the existence of a strong couplingbetween mechanical, thermal and diffusion phenomena.

From a modelling viewpoint, the theoretical description ofthese coupling phenomena is generally based on an empiricalor thermodynamic interpretation of experiments. These inter-actions are taken into account at two levels: in the expressionof polymer characteristics as a function of pressure, temper-ature and concentration on the one hand, in the thermo-dynamic formulation of the model on the other hand, thatcorrespond to the indirect and direct couplings respectively.

The indirect couplings were specially considered inthe literature. A number of molecular models, based onthe polymer internal structure, represent the diffusioncharacteristics variation with pressure, concentration and/ortemperature. Some bibliographical syntheses are given onthis subject by Liatsis (1989), Gaillard-Devaux (1995) andmore recently by Klopffer and Flaconnèche (2001). In thesame way, there is some information on mechanical andthermal characteristics. For instance, Rodier-Renaud (1994)define the isobaric expansion coefficient as a function oftemperature for polyethylene. Nevertheless, the complexityof mechanisms, the multiplicity of dependences and thedifficulty in carrying out characterization tests under high gaspressure do not make it possible to define all the indirectcouplings and often lead to complicated expressions of thesecouplings with limited validity domain.

The direct couplings were tackled in the classicalframework of thermodynamics of irreversible processes.They are taken into account via a direct reliance ofthermodynamic potentials (specific free energy, dissipationpotential) on all the variables (mechanical, thermal,diffusion). This second approach of couplings enables one torepresent successfully a number of physical phenomena. Togive just one example, it is easy for Soret effect (i.e. the case

in which a thermal gradient induces a mass flux) to bedescribed (Papon and Leblond, 1990). Cailletaud et al.(2002)presented a bibliographical work on this direct way ofconsidering couplings. In this paper, metallic materialbehaviour has been studied, especially the interaction betweenmetal oxidation and mechanical load. In polymer mechanics,interesting theoretical developments, which led to the DNLR(distributions of non-linear relaxations) model, were alsoproposed by Cunat (1991) and fellow workers. A distributionof relaxation processes is used to represent the internaldissipation phenomena. At any time, the system tends towardsa state of equilibrium in accordance with some physicalconsiderations. This formulation makes it possible to takeaccount of a lot of direct couplings (mechanical, magnetic,thermal, etc.). While this modelling thermodynamicframework is well established, nevertheless the relaxationspectra definition is delicate when couplings are present andwas not considered in the explosive decompression case. So,actually there is no clearly defined model in which thesedirect couplings are thoroughly considered.

Faced with this situation, a modelling which takes thedirect couplings as well as the indirect couplings into accountshould be developed in the long term. It is a prerequisitewhen one aims at foreseeing the damage incipience. The firststep, which is the subject of this article, is to establish amodel in which only the direct couplings are considered. Thematerial is considered at the macroscopic scale in theframework of thermodynamics of continuous media withinternal variables.

In the first part of this document, we propose some ideason the elementary representative volume (ERV) description.Five alternatives are considered. They differ in the way ofdescribing the polymer internal structure on the one hand, andin the way of describing the gas transport phenomenon on theother hand. Then, the simplest one is taken as a starting pointof thermodiffuso-mechanical modelling, in other words theERV is considered to be a homogenous polymer and gasmixture. Gas diffusion is a molecular diffusion, which obeysa Fick’s type law and thus is driven by concentrationgradients. We look for the easiest approach to the directcouplings that is why a classical thermodynamic (formechanic) framework is chosen. The thermodynamicpotentials employed are quadratic forms so that the equationsof state as well as the complementary evolution laws arelinear. This model does not account for the indirect couplings,which means that material characteristics have constantvalues. The second part of this article deals with a particularversion namely the linear thermodiffuso-elastic model.

Its numerical implementation in ABAQUS™ led to a firstqualitative study of the direct couplings. The structure underconsideration, corresponding to an infinite pipe, wassubjected to mechanical, thermal and diffusion loads. Theeffects of coupling on its elastic response are presented in thethird part of the document.

573


Firstly, this numerical tool will enable one to estimate thecontribution made by such a modelling to the explosivedecompression study. Admittedly this modelling is limited,but such a tool taking account of the interaction betweenmechanics, thermic and diffusion will facilitate thecomprehension of the basic phenomena. It is an obligatorystage if one wishes, in the long run, to approach damagephenomena. Secondly, the simulations will lead to a newprotocol of experimental characterizations, which will in thelong term make it possible to distinguish the effect of thedirect couplings from the effect of indirect couplings due tothe intrinsic dependences of polymer characteristics.

1 CHARACTERIZATION OF THE ERV

In this development, the thermodiffuso-mechanical behav-iour of polymers is described at a macroscopic scale in theframework of the thermodynamics of continuous media withinternal variables. The physical “poverty” of this modellingis filled in by a more or less complicated kinematics, whichleads to a relevant representation of the microstructure. AnyERV is represented by a mono- or biphasic entity, in whichgas transport is due to the concentration and/or pressure gra-dients. So, five alternatives were considered, correspondingto an increasingly elaborate approach to the strain internalmechanisms. Interest here is in the various modelling whichcould be developed within the framework of polymermechanics.

1.1 ERV Description Possibilities

1.1.1 Monophasic Modelling

First of all, the polymer is described as a monophasicmedium, which is particularly appropriate to amorphousmaterials such as thermosetting elastomers and uncrystallizedthermoplastics. The gas transport corresponds to a moleculardiffusion and/or a flow. These different perceptions oftransport phenomena lead to the three models, which arepresented below.

Homogeneous Mixture

In the first model, gas diffuses within the material accordingto a Fick’s type law, that is to say under the effect ofthe concentration gradients. Then, the ERV is described asa classical continuous medium, made up of a homogeneouspolymer-gas mixture. The equations of state as well asthe complementary evolution laws relate to this mixture(Fig. 1).

Damage phenomena can be approached only with internalvariables, which have to be numerous enough to accountfor the damage modes observed during an explosivedecompression.

Saturated Porous Medium with an Impermeable Skeleton

In the second model, gas flows within the material accordingto a Darcy’s type law, i.e. under the effect of pressuregradients. So, the ERV is considered as a saturated porousmedium in which the polymer corresponds to the imper-meable skeleton and the gas saturates the connected porousspace (Fig. 2).

In this case, damage processes can be taken into accountclassically by some internal variables but also by amechanical effect of gas-polymer interaction correspondingto the internal structure effect (pre-existing defects). Thus,this second ERV representation contains more potentialitiesfor the modelling of damage than the first one. It is adifferent physical description, which employs some geo-metrical variables associated with a microscopic scale (i.e.porosity and tortuosity). The presence of these interconnectedcavities is debatable at a mesoscopic scale because they arenot observable. However, it is conceivable at a lower scale.One can imagine that gas molecules contained in thesecavities are numerous enough to be governed by classicalequations of state (PVT) as in a porous medium. Somequestions remain about interconnectivity, tortuosity and freevolume notion. Besides, the treatment of the initial state (i.e.before absorption) poses also a problem. As the “void” is notconceivable in the connected porous space, the nature of thegas, which is present at the initial state, has to be definedfrom physical considerations.

574

Figure 1

Homogeneous mixture.

Figure 2

Saturated porous medium with an impermeable skeleton.

Gas - Fick

Polymer - Fick

Polymer

Gas - Darcy


Such a modelling was already considered in the literature.For instance, a thermoporo-elastic model was proposed byWeiler (1992) to study the rocket ablative materials such ascarbon-phenolics. The formulation suggested by Coussy(1991) was also applied in the context of the thermoporo-elastic behaviour of rocks (Onaisi, 1991) and the thermo-hydro-mechanical one (Skoczylas et al., 1992). All thesestudies highlighted some effects of coupling.

Saturated Porous Medium with a Permeable SkeletonIn the third model, the gas transport corresponds to a mole-cular diffusion within the skeleton (i.e. according to a Fick’stype law) and also a flow within the connected porous space(i.e. according to a Darcy’s type law). The ERV is describedas in the previous case, except that now the polymer, whichforms the skeleton, is permeable to the gas (Fig. 3).

Figure 3

Saturated porous medium with a permeable skeleton.

The gas is absorbed in the skeleton on time scales whichare different from those associated to the outflow within theconnected porous space. That is why this modelling candescribe the hysteresis effects and non-linearities observedduring the diffusion tests. Moreover, this approach makes itpossible to account for the complicated evolutions of thepolymer characteristics with impregnation. Finally, differentdamage mechanisms can be considered according to theinteraction between gas and polymer.

1.1.2 Biphasic Modelling

In the framework of semi-crystalline polymers, the ERVrepresentation can be improved by taking account ofthe polymer morphology. Cangémi and Meimon (2001)proposed an advanced mechanical model in which thepolymer is described as a biphasic or porous medium. Eachconstituent has its own constitutive law. The spherulites andthe constrained amorphous phase form the skeleton whereasthe free amorphous phase defines the “fluid”. This last flowswithin the connected porous space of each ERV. Thismodelling led to a satisfactory representation of the polymermechanical behaviour.

The introduction of diffusion and coupling requires furtherdevelopments. Studies made by the IFP under gaseouspressure corresponding to an explosive decompression,showed that the gas diffused mainly in the free amorphousphase. Nevertheless, it is very probable that under higherpressure, gas diffuses also within the constrained amorphousphase associated to the spherulites. Then, two alternativesare considered.

Unsaturated Porous Medium with an Impermeable SkeletonIn the fourth model, gas diffuses within the free amorphousphase according to a Fick’s type law and the homogenousmixture of gas and free amorphous phase flows within theconnected porous space, governed by a Darcy’s type law.

The ERV is still described as a porous medium filled nowwith the mixture of gas and free amorphous phase (Fig. 4).So, the problem of the initial state (i.e. before absorption)does not arise any more. This representation containsinteresting morphological characteristics. Even if thedescription of the polymer microstructure is poor, it offersgreat potentialities in the differentiation of damage modes.

Unsaturated Porous Medium with a Permeable SkeletonIn this last model, the gas diffuses also between the crystallinelamellae (i.e. in the constrained amorphous phase linked to thespherulites) according to a Fick’s type law. Indeed, theskeleton is assumed to be permeable to the gas. Each compo-nent of the mixture has to be considered separately (Fig. 5).

Polymer

Gas - Fick

Gas - Darcy

575

Figure 4

Unsaturated porous medium with an impermeable skeleton.

Figure 5

Unsaturated porous medium with an permeable skeleton.

Polymer: spherulitesand constrainedamorphous phase

Polymer: freeamorphous phase

Gas in free amorphousphase - Fick

Mixture of gas and freeamorphous phase- Darcy

Polymer: spherulitesand constrainedamorphous phase

Polymer: freeamorphous phase

Gas in free amorphousphase - Fick

Mixture of gas and freeamorphous phase- Darcy

Gas in the skeletonformed by spherulitesand constrainedamorphous phase - Fick


Such a modelling will probably describe some strong non-linearities and will constitute an improvement in the damagetreatment as well as in the approach of the impregnationeffects.

1.2 Modelling Choice

Increasingly complicated descriptions of the ERV have beenproposed. If couplings were specially considered in theframework of the porous media, it is less usual in the case ofmixtures framework. Besides, the semi-crystalline polymeris described as a porous medium in which the “fluid”corresponds to a mixture of free amorphous phase and gas.

Thus, the mixtures framework is chosen to develop thethermodiffuso-mechanical model. Such a modelling willenable one to approach the amorphous polymer behavioursubjected to an explosive decompression. Moreover, it is aprerequisite work when one aims at modelling the semi-crystalline behaviour.

2 THERMODIFFUSO-ELASTIC MODELLING

2.1 Thermodynamic Framework

The model is developed in the framework of the generalizedstandard media and therefore follows a classical approach. Afield of state variables represents the material at any time.The introduction of a first potential (specific free energy)makes it possible to set laws linking thermodynamic forces(stress, entropy, chemical potential) to dual variables (strain,temperature, normalized concentration). The dissipations,which are only associated to thermal and diffusionphenomena in an elastic behaviour case, are discussed usinga second potential (dissipation potential). This potentialexpression depends on the considered coupling assumption(total or not). Once defined, it leads to complementaryevolution laws (Fourier’s, Fick’s). These different steps arepresented in the following paragraphs.

In this thermodynamic framework, every ERV is describedat a macroscopic scale as a bicomponent, continuous,homogenous medium: polymer and gas. One focuses on theaverage behaviour of the mixture defined by average physicalvalue (based on each species value). Moreover, this ERV isopen and thus trades heat as well as matter with thesurrounding medium. The two-species diffusion is amolecular diffusion, which obeys a Fick’s type law and thenis driven by concentration gradients. In addition, the materialis assumed to be isotropic and is studied in the framework ofsmall perturbations. Finally, a strong assumption consists inkeeping constant each ERV characteristic in time and space.This approximation is by no means verified but makes itpossible to simplify strongly the equations on the one handand to account for only the direct couplings on the other hand.

2.2 Local Equations

The behaviour of polymers subjected to pressure, gasconcentration as well as temperature variations is defined bydiffusion, dynamic and thermal equations. The two formerequation families are classically developed but the thermalone is not so easy since it contains couplings correspondingto dissipative phenomena.

2.2.1 Mass Balance

In a bicomponent system without chemical reaction, the massof each species remains separately the same, which leads totwo mass balance equations.

The two components of the mixture are characterized bythe following variables:– ρ average mixture density (kg/m3);– ci normalized concentration of the i th constituent (Pa),

defined as the concentration (mass fraction, i.e. kg/kg)and solubility si ratio (Pa–1);

– J→

mirelative mass flux of the ith diffusing phase (kg/m2·s).

The mass balance for the i th component leads to thefollowing diffusion equation:

This classical balance equation is applied to polymer andgas. Nevertheless, mass flux and normalized concentrationssatisfy by definition the following equations:

So, both the diffusion equations are linked and thus onlythe gas one will be used further in this paper:

(1)

2.2.2 Mechanical Balance

The mechanical balance equation is derived from the funda-mental principle of dynamics. When one neglects the inertiaeffects (ργ→≈ 0

→), it leads to the following static equation:

(2)

where σ= is the total Cauchy stress tensor (Pa), f→

is the bodyforce per unit of mass (N/kg) at any point within the ERVand γ→

the considered particle acceleration (m/s2).

2.2.3 Thermodynamic Balance – First and Second Laws

Thermodynamic Potential, Equations of StateIn the context of thermodynamics of continuous media, oneclassically employs the specific free energy ψ of the mixture(J/kg). For an elastic behaviour, this potential depends onfour state variables: the temperature T (K), the infinitesimal

r rdivσ ρ+ f

ρsc

tJg

gmg

d

ddiv= −

r

r rJ J s

c

ts

c

tm m gg

pp

g p= − =− and

d

d

d

d

ρs

c

tJi

imi

d

ddiv= −

r

576


strain tensor ε=e and the gas and polymer normalized

concentrations cg and cp, respectively:

ψ = ψ (ε=e, T, cg, cp)

When a normal configuration is assumed and the first twothermodynamics laws are considered, constitutive laws alsoknown as equations of state can be derived from ψ as follows:

where s (J/kg·K) denotes the specific entropy and µi (J/kg)the mass chemical potential for the ith species (dual variableof the mass fraction).

Remark: means partial derivation with ★ as

constant value.Under the assumption of small perturbations, that is to say

infinitely small strains and small differences of temperatureand normalized concentration, the development of thepotential ψ to the second order leads to the following linearconstitutive equations:

(3)

(4)

(5)

where:– T0, c0

g , σ= 0, s0 and µ0i respectively represent initial

temperature, gas normalized concentration, stress, specificentropy and chemical potential of the ith constituent at zerostrain, thermal and concentration variations;

– is the specific heat at constant strain

and normalized concentration (J/kg·K);– λ and µare the Lamé elastic coefficients (Pa);– αT (K) and αc the isotropic expansion coefficients linked to

respectively heat conduction and mass transport;– the effect of temperature variation (concentration variation)

on the chemical potential (entropy) is linked to the coef-ficient d (J/kg·K);

– D is defined as a function of coefficients ρ, D and kµ, thetwo latter ones being defined further in the document(m2/s2);

– I= is the identity matrix.

Remark: tr represents the trace operator.

Heat EquationLet J

→

q (J/m2·s) be the heat flux per unit area and r (J/m3·s)the volume density of heat generated by an outer source (e.g.by an radiating source), the heat balance equation takingaccount of the different couplings has the followingexpression:

(6)

The last term corresponds to the temperature variationlinked to gas diffusion within the polymer material.

DissipationsThe second law of thermodynamics makes it possible todefine the volume dissipation Φ (positive or nil) linked toentropy generation. An elastic behaviour is characterized bythe absence of internal variables, which leads to a nil intrinsicdissipation. Φ splits into two terms, Φ1 and Φ2 (J/m3·s),which are linked respectively to the temperature andchemical potential gradients, so that:

where:

is the entropy flux (J/s·K·m2).Introduction of the dissipation potential d and its dual d*

—through Legendre-Fenchel transformation— (J/m3·s) iscompatible a priori with the second law of thermodynamics.The expressions of these continuous, convex and positivefunctions depend on the considered coupling assumptions.Evolution equations are given for different d* in thefollowing paragraph.

Coupling ProcessingUnder the assumption of a total coupling between dissipativephenomena, the dual potential becomes a unique function,which is given by Equation (7). In the framework ofgeneralized standard media (i.e.normal dissipativity assump-tion), one sets the evolution complementary equations givenby Equation (8).

(7)

(8)

r rJ

d

TJ

ds m

g pTg p

g= ∂

∂ −[ ]

= ∂

∂ − −( )[ ]

∗

− −( )

∗

−grad grad

grad gradµ µµ µ

,

d d T g p∗ ∗= − − −[ ]( )grad grad, µ µ

r r rJ

TJ Js q g p mg

= − −( )[ ]1 µ µ

Φ = − ⋅ −( )2

rJm g pg

and gradµ µ

Φ Φ Φ Φ= + ≥ = − ⋅1 2 10,

rJ Ts grad

µ µ ρT sc

tg p ggd

d− −( ) +[ ]d

ρ λ µ α

εεC

T

tJ r T

te

ic q T

e

,

d

ddiv tr

d

d= − + − +( )

r3 2

C TT

ei e

i

cc

εε

,,

= ∂∂

s

ρλ µ α εc

e T T− +( ) − −( )01

3 2 dtr

µ µ µ µg p g p g g gs c c− = −( ) + −( )0 0 0D

d+ +( ) + −( )013 2 s c cT

eg g gρ

λ µ α εtr

s s = + −( )00

01

TC T Te

icε ,

λ µ α α− +( ) −( ) + −( )[ ]003 2 T c g g gT T s c c I

σ σ λ ε µε= + ( ) +0 2tr e eI

∂∂

★

σ ρε

µε ε

= ∂∂

= − ∂

∂

= ∂

∂

≠

Ψ Ψ Ψe

T c c c ci

i i T cg p

eg p

ek i

T s c, , , , , ,

,s and 1

577


Moreover, when the dissipation phenomena are assumed tobe driven by linear and fully coupled laws, the potential d* isquadratic and leads in an isotropic case to the following laws:

where:– kµ is linked to the chemical potential gradient effect on the

gas mass flux (kg·s/m3);– kT represents the thermal gradient effect on the entropy

flux (W/m·K2);– cTµ is a coupling coefficient corresponding to the tem-

perature (chemical potential) gradient effect on the mass(entropy) flux (kg/m·s·K).With the previous expression of the chemical potential

(Eq. 5), one obtains:

When dissipative processes are assumed totally uncoupled,two dual potentials d1

*, d2*, enter the expression of d*:

As previously stated, an important simplification of theseequations consists in considering the dual dissipation potential

as a quadratic form. One then obtains the well-knownevolution equations, namely Fourier’s (only conduction case)and Fick’s (diffusion case) equations:

where Λ is the thermal conductivity (J/m·s·K) and D thediffusivity (m2/s).

Summary: Unknowns and Constitutive EquationsThe variables defining the thermodiffuso-mechanical state ofthe material (gas-polymer mixture) are the components ofthe displacement vector, the temperature T and the gasnormalized concentration cg. Among all the different coef-ficients that were introduced, kµ, cTµ and d are the only onesstill to be determined through a parametric study.

With the previous equations, the thermodiffuso-elasticmodel is finally defined under a full coupling assumption asfollows:– Mechanical problem:

(9)

– Diffusion problem:

(10)

– Thermal problem (Eq. 11):

λ µ αρ

εµ µ µdiv grad tr div gradk c k Tc eT

+( ) [ ]( ) + −( ) ( )3 2d

ρ ρ10d

d10 div grad6 6s

c

ts cg

gg g= ( )D

λ µ α α− +( ) −( ) + −( )[ ]06 03 2 10T c g g gT T s c c I

σ σ λ ε µε= + ( ) +0 2tr e eI

r r rdivσ ρ+ =f 0

rJ k s c

km g g= − =and grad

g,µ

µρD D

D

rJ k T T k

Tq T T= − =grad ,Λ

d d T d g p∗ ∗ ∗= −( ) + − −[ ]( )1 2grad gradµ µ

++( ) + −( )[ ] [ ]λ µ α

ρµ µ εµ µ grad trc T kc

T g pe3 2

− + −( )[ ]Dρ µ µµ

µ µkc T k s cT g p g ggrad

rJ k c T c k Tq T T T g p= − −( ) + −( ) −( )[ ]d dµ µ µ µ µ grad

k k c Tc eTgrad tr grad+

+( ) [ ] + −( )µ µ µλ µ α

ρε d

3 2

rJ k s cm g gg

grad= − µ D

rJ k T cs T T g p= − − −[ ]grad gradµ µ µ

rJ k c Tm g p Tg

= − −[ ] −µ µµ µgrad grad

578

ρ

ρ λ µ αε

ρλ µ α

ρε

ε µ µ µ µ

µ

CT

t Tk c T T r k c T

ks c T

t

s

egY T T

g g T

e

gc e

,

d

d div grad grad

10 grad trd

d

10 grad tr

6

6

= + −( )

( ) + + −( ) ( )

+ ( ) ( ) − +( )

−+( )

Λd d d d2 2

13 2

23 2

2

2 2D

D [[ ] ⋅ ++( )

[ ]( )

+ −[ ] ( )− −[ ] +( ) [ ]( )+ −[ ] ⋅

grad grad tr

10 div grad

div grad tr

10 grad

6

6

c k

c kk

s T c

c k T

c kk

s c

gc e

T g g

Tc e

T g g

µ

µ µµ

µ µ

µ µµ

λ µ αρ

ε

ρ

λ µ αρ

ε

ρ

3 2

1

3 2

21

22

d

d

d

D

D gradgrad

grad tr grad

T

c k TTc e− −[ ] +( ) [ ] ⋅2

3 2µ µ

λ µ αρ

εd

(11)

G Rambert et al./ A Modelling of the Coupled Thermodiffuso-Elastic linear Behaviour 579

3 QUALITATIVE STUDY OF COUPLING

The numerical implementation of the linear thermodiffuso-elastic model in the software ABAQUS™ leads to a firstqualitative study of direct couplings. The programmedelements are first validated considering an uncoupledbehaviour. Then, couplings are introduced gradually and theireffects on the polymer sheath of an infinite tube are given.The model contains five coefficients that define a more or lesscomplicated coupling between mechanical, thermal anddiffusion phenomena: αT, αc, kµ, cTµ and d. The kµ parameteris studied first, because it cannot be eliminated. Secondly, themore classical coefficients of expansion, αT and αc, areintroduced. cTµ and dparameters are considered finally.

3.1 Structure and Loading

The structure considered is a flexible pipe used for carbondioxide transportation and formed by a polymer sheath. Itsdimensions as well as the loading path and the boundaryconditions are chosen so that the assumption of plane straincan be considered. Due to symmetries, only one quarter ofthe pipe section is represented by 3000 8-node quadrilaterals(Fig. 6).

All except the three unknown coupling coefficients(kµ, cTµ and d) correspond to the polyvinylidene fluoride(PVDF) characteristics. They are measured at (Tie Bi, 2000)or extrapolated to (Flaconnèche et al., 2001) a temperaturearound 21°C (Table 1). The coefficient of expansion due togas diffusion is determined at 130°C (data issued fromrecent tests, IFP). It is a relatively rare data in the literaturebecause it requires specific experimental systems. Never-theless, such an approximation of the characteristics valuescan be admitted because of the exploratory nature (in someaspects) of this study. All these material characteristics areassumed to be constant during the simulation.

The loading path is defined as corresponding approxi-mately to the engineering applications (Fig. 7). However, thethermal exchange between the transported fluid and thesheath is not taken into account, in particular during theexplosive decompression, which induces a temperature dropinside the pipe and therefore a modification of the insideboundary conditions.

3.2 Uncoupled Case

In the uncoupled case, four of the five coupling parametersare equal to zero:

αT = 0, αc = 0, cTµ = 0 and d = 0

The following simplified system of equations describesthe problem:– Mechanical problem:

(12)

– Diffusion problem:

(13)

– Thermal problem:

(14)

As the coefficient kµ appears in the denominator of the lastterm of Equation (14), it cannot be cancelled. So, it is chosenin such a manner that the effect of diffusion on thermalproblem can be neglected. Specifically, for a kµ around1030 kg·s/m3, one obtains the same results that those issuedfrom the uncoupled mechanical, thermal and diffusionsimulations of ABAQUS™. In other words, this value of kµcan be considered as relevant to an uncoupled case.

3.3 Diffusion Effect on Thermal Problem via kµ

kµ appears above to be the only parameter necessarilydifferent from zero. It is possible to make its effect negligiblebut not to suppress it. That is why it is the first oneintroduced in the model.

Theoretical AspectsFrom a thermodynamic viewpoint, kµ defined as an Onsagercoefficient is necessarily positive. Some simulations aremade for the following parameters set:

0 < kµ ≤ 1030, αT = 0, αc = 0, cTµ = 0 and d= 0

In these conditions, the thermodiffuso-elastic model ischaracterized by the foregoing equations (Eqs. 12, 13, 14).Strictly speaking, taking account of kµ only, mechanical aswell as diffusion problems correspond to the truly uncoupled

ρ ρε

µC

T

tT r

ks ce

gY g g,

d

ddiv grad 10 grad6= ( ) + + ( ) ( )Λ 1 2 2

D

d

ddiv grad

c

tcg

g= ( )D

r r rdiv with trσ ρ σ σ λ ε µε+ = = + ( ) +f Ie e0 20

TABLE 1

Characteristics of a PVDF with CO2 at 21°C(except αc which is measured at 130°C)

ρ E ν Cε=e,Y

gΛ αT sg D αc

(kg/m3) (MPa) (∅) (J/kg·K) (W/m·K) (K –1) (Pa–1) (m2/s) (∅)

1745 1743 0.38 1045 0.26 10–5 5.539·10–3 9.8·10–12 2.2·10–13


5 mm 1 mm

Polymer sheath

Outside side

T = 21°C

Inside side

Jq = 0 /Jmg = 0

cg = 0

Jq = 0

Jmg = 0

Figure 6

Section and meshing of the pipe.

00

10

1800

P (

MP

a)

37 800 37 890 75 690 Time (s)

00

10

1800

c g (

MP

a)

37 800 37 890 75 690 Time (s)

021

150

1800

T (

°C)

75 690 Time (s)

Temperatureinitially at 21°C in all the structure

Gas normalized concentrationinitially at zero in all the structure

Pressure

Figure 7

Boundary conditions and loading applied on the inside face of the pipe.


case. The sole heat equation is modified: diffusion pheno-menon induces now temperature variations. Thus, the onlythermal response is studied.

Numerical AspectsThe effect of diffusion on thermal problem results in a localincrease in temperature. This result was foreseeable becausethe coefficient Kµ relative to this effect is superior (or equal)to zero (Eq. 14).

The foregoing effect is negligible when the coefficient kµis between 1030 and 5·106 kg·s/m3. Indeed, the relativevariation of temperature compared to the uncoupled case isthen lower to one percent. When the value of kµ decreasesfurther (below 5·106 kg·s/m3), the coupling effect appears.The temperature rise is more important as kµ is small, a resultconsistent with the heat equation (kµ appears to thedenominator, Eq. 14). Moreover, a second threshold valueoccurs around 2·104 kg·s/m3: an inferior kµ leads todisproportionate coupling effects with a relative variationof temperature around a hundred percent during thedecompression. In the interval [2·104 kg·s/m3, 5·106 kg·s/m3],this relative variation compared to the uncoupled situationremains more realistic, namely inferior to fifty percent.

To estimate this effect of diffusion on the thermal field,elements placed on a tube radius are followed during thesimulation. Whatever the kµ value, the maximal relativescatter of temperature is recorded at roughly 0.27 and0.6 mm from the interior side during the compression anddecompression phases, respectively (Fig. 8). This result isclosely linked to the loading and boundary conditionscorresponding to this simulation (Figs. 6 and 7).

During the first stage of the simulation (linearly increasingand constant concentration steps), the relative variation oftemperature compared to the uncoupled case increases thendecreases at a given point (Fig. 8). It reaches a maximalvalue faster at a point located nearer from the interiorboundary than another one. This phenomenon can beexplained by the evolution of the gas mass flux in space andtime, in other words it is connected with the characteristictime of diffusion process. Moreover, a zone characterized bymore complicated fluctuations of temperature appears on theradius owing to the nil concentration imposed on the outsideedge of the structure. Indeed, this boundary condition resultsin strong gradients of concentration near the outside edge atan advanced stage in the diffusion phenomenon. For someparticular values of kµ, the stationary state is not reachedcompletely when decompression occurs.

During the second stage of the simulation (linearlydecreasing and zero concentration steps), the heat increase isquite steady. The relative variation of temperature is maximalat the end of the linear unloading and then it tends towardszero (Fig. 8). Indeed, concentration gradients disappear as anil concentration is imposed on each edge.

The trace of temperature curves at two points of a radius(Fig. 9) leads to the same observations.

Physical AspectsThis first “rudimentary” coupling describes the effect ofdiffusion on temperature evolution. Whatever the direction ofdiffusion is, the gas transport results in a local increase in thepolymer temperature via kµ. From a physical viewpoint, sucha temperature rise results from the evolution of polymerinternal state, whose gas content varies with the pressureapplied on the boundary.

3.4 Introduction of Expansions

The two other coefficients introduced represent the expansionlinked to the variations of temperature (αT) on the one handand of concentration (αc) on the other hand.

3.4.1 Thermal Expansion

First, only the thermal expansion coefficient is considered,which leads to the following configuration:

104 ≤ kµ ≤ 1030, αT ≠ 0, αc = 0, cTµ = 0 and d = 0

Theoretical AspectsMechanical and thermal problems are thus modified. In thefirst one, thermal dilation occurs (Eq. 15)and in the secondone, the rate of volumetric swelling strain induces avariation of temperature (Eq. 16). The diffusion equationremains the same as in the previous case (Eq. 13);nevertheless gas diffusion has an indirect effect onmechanics through the temperature field (Eq. 16). Indeed,diffusion acts on the temperature evolution via kµ, which hasfurther repercussions on thermal expansion and thus on themechanical problem. This last modifies finally the thermalproblem through the variation of the rate of volumetricswelling strain.

(15)

(16)

E (Pa) and ν being the coefficients of Young and Poisson,respectively.

Numerical AspectsAs the processes are relatively slow during the simulation,the temperature field is not affected by this classical coupling(which needs faster solicitations to be active).

The indirect effect of diffusion on mechanics results in anonlinear distribution of volumetric swelling strain along

ρ λ µ αε

µks c T

tg g T

e

10 grad trd

d6+ ( ) ( ) − +( )1

3 22 2

D

ρ εCT

tT re

gY,

d

ddiv grad= ( ) +Λ

ε ε ν σ ν σ αeTI T T I= + + − ( ) + −( )0 0

1

E Etr


0

25

50

5 5.25 5.5 5.75 6Radius (mm)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

0

25

50

5 5.25 5.5 5.75 6Radius (mm)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

t = 900 s

t = 5400 s

t = 37 800 s

t = 1800 s

t = 9000 s

t = 2520 s

t = 19 800 s

t = 37 810 s

t = 38 070 s

t = 43 290 s

t = 37 870 s

t = 38 790 s

t = 37 890 s

t = 40 410 s

t = 47 970 s t = 59 490 s

Linearloading step

Constant stepLinear

unloading stepConstant step

0 900 1800 2520 5400 9000 19 800 37 800 37 810 37 870 37 890 38 070 38 790 40 410 43 290 47 970 59 490 Time (s)

Figure 8

Spatial distribution of the relative scatter of temperature ([T – T1] / |T1|), for the coupled case (T) compared with the uncoupled case (T1) forkµ = 2·104 kg·s/m3 at different time.

20

180

220

0 40 000Time (s)

Tem

pera

ture

(%

)

140

100

60

20 000

r = 5.25 mm r = 5.75 mm

1E4

1E5

5E4

1E30

2E4

5E5

20

80

100

0 40 000 60 00060 000Time (s)

Tem

pera

ture

(%

)

60

40

20 000

1E4

1E5

5E4

1E30

2E4

5E5

0

5

6

r (mm)

Figure 9

Temperature variation at two points of the radius for different kµ.


a radius (Fig. 10). The profiles of thermal expansion givenfor a kµ coefficient equal to 2·104 kg·s/m3 (coupled case) and1030 kg·s/m3 (uncoupled case) show this phenomenon.Moreover, during the transient phases, a slight peak of strainappears near the interior edge of the structure. So, accordingto geometry and boundary conditions, one could eventuallyexpect a localization of stress generated by couplingphenomena.

3.4.2 Expansion Linked to Gas Diffusion

At this step, the expansion coefficient αc corresponding tothe gas diffusion is introduced in the model:

104 < kµ ≤ 1030, αT ≠ 0, αc ≠ 0, cTµ = 0 and d= 0

Theoretical AspectsThe problem is modified overall. In this case, the couplingbetween thermal and diffusion phenomena via kµ is inevi-table. Indeed, when a non-zero coefficient αc (coefficientexpansion due to diffusion) is considered, any plausible valueof kµ favours this effect of diffusion on temperatureevolution. So, the basic modelused to describe the thermo-diffuso-mechanical behaviour of polymers contains neces-sarily all the couplings considered in this paragraph.

The values of kµ are chosen so that the concentrationevolution is governed mainly by the classical term linked tothe divergence of the concentration gradient. This conditionleads to:

104 < kµ ≤ 1010

0.E-00

4.E-03

5.E-03

5 5.25 5.5 5.75 6Radius (mm)

The

rmal

exp

ansi

on

3.E-03

2.E-03

1.E-03

Linearloading step

Constant stepLinear


0 180 540 900 1260 1800 2520 9000 37 800 37 810 37 870 37 890 38 070 38 790 40 410 43 290 75 690 Time (s)

t = 540 s

t = 1800 s

t = 37 800 s

t = 900 s

t = 2520 s

t = 180 s

t = 1260 s

t = 9000 s

2.E4

0.E-00

4.E-03

5.E-03

5 5.25 5.5 5.75 6Radius (mm)

The

rmal

exp

ansi

on

3.E-03

2.E-03

1.E-03

t = 37 870 s

t = 38 790 s

t = 75 690 s

t = 37 890 s

t = 40 410 s

t = 37 810 s

t = 38 070 s

t = 43 290 s

2.E4

0.E-00

4.E-03

5.E-03

5 5.25 5.5 5.75 6Radius (mm)

The

rmal

exp

ansi

on

3.E-03

2.E-03

1.E-03

t = 540 s

t = 1800 s

t = 37 800 s

t = 900 s

t = 2520 s

t = 180 s

t = 1260 s

t = 9000 s

1.E30 1.E30

0.E-00

4.E-03

5.E-03

5 5.25 5.5 5.75 6Radius (mm)

The

rmal

exp

ansi

on

3.E-03

2.E-03

1.E-03

t = 37 870 s

t = 38 790 s

t = 75 690 s

t = 37 890 s

t = 40 410 s

t = 37 810 s

t = 38 070 s

t = 43 290 s

Figure 10

Profiles of thermal expansion for kµ = 2·104 kg·s/m3 and kµ = 1030 kg·s/m3.


Numerical AspectsWhatever the value of kµ, the thermal results are identical tothose obtained in the previous case. A weak slowing downof the mass transport is observed. Moreover, the radialdisplacement, which corresponds to a decrease in volume(Fig. 12) in the uncoupled case, is reversed. A competitionappears between internal pressure that compacts the sheathand diffusion as well as thermal processes that inducean expansion of the structure. The thermal (Fig. 10) anddiffusion (Fig. 11) expansions have different orders ofmagnitude, but it is important to remind the measurementuncertainty of αc and αT.

3.4.3 Physical Aspects

With these two expansion coefficients, the basic exper-imental results observed by IFP are recovered (contractionunder the pressure effect—expansion under the absorptioneffect). With the aim of differentiating the two effects, it will benecessary to know the local value of temperature. Theseeffects are expected to be quantified during a classicalpermeability test. After a transient phase, the gas mass fluxbecomes stationary and a thermal gradient appears in thethickness owing to the “kµ” coupling. Then, expansionslinked to the variations of temperature and concentrationinduce a non-linear variation of sample thickness. A compa-rison between numerical predictions and local measures hasto be considered.

3.5 Introduction of cTµ

Theoretical AspectsSome supplementary terms appear in diffusion and thermalproblems. The divergence of the thermal gradient contributesto the concentration variation. In order to conserve classicaldiffusion and conduction effects, kµ and cTµ are defined asfollows:

104 < kµ ≤ 1010 and 0 <cTµ < 107

Numerical AspectsThe effects of cTµ are similar whatever the value of kµ butthey are all the more strong as kµ is small.

For small values of cTµ (10–2 kg/m·s·K), gas diffuses alittle bit more slowly during the linear loading step (Fig. 13,negative absolute difference of concentration) and a little bitmore quickly during the decompression step. Thus, thiscoefficient plays a directional role in diffusion process(absorption or desorption). It is a diminishing and/or en-hancing factor. Its effect on temperature field has the sameorder of magnitude. During the loading step, temperature ismore elevated. During the decompression step correspondingto a constant temperature on the boundary, a weak drop intemperature appears inside the material due to the accel-eration of gas desorption.

For high values of cTµ (103 kg/m·s·K), diffusion isaccelerated by the cTµ div(grad T) term which appears in theconcentration evolution equation. Diffusion and conductionhave the same characteristic time (Fig. 14). Concentrationbecomes equal to zero at the end of decompression step.Temperature increases more slowly during the two first steps.During the decompression, temperature field corresponds toa linear distribution along the radius; this is linked to theboundary conditions of temperature.

Physical AspectsIt becomes more difficult to explain the coupling effects atthis stage. The new coefficient cTµ appears as a multiplier inof some gradients. If within the structure considered, loadingand boundary conditions generate strong and/or localisedgradients, the effect of cTµ will probably be more significant.This coefficient acts on the kinetics of diffusion andconduction (more or less rapid) with constant D and Λ.

3.6 Totally Coupled Case

Now, all the coupling coefficients are considered. The studyof constitutive equations leads to three threshold values of d:

Numerical AspectsSome problems of convergence appear often during thesesimulations. They could be suppressed by modifying theevolution of time step. Two results are presented in this partwith the aim of underlining the effects of d.

First, d is chosen equal to d1 and the results are comparedto a case without cTµ and d. The thermal problem is modifiedand, via the thermal expansion, also the mechanical one.Some difference of concentration appears in several localizedparts of the structure. The variation of temperature is all themore important as the coefficient [c2

Tµ / kµ] is close to [l/T].The effect of d on each problem is then studied. For a

given couple (kµ, cTµ), simulations are done with differentvalues of d. It is difficult to associate a global effect to thiscoefficient because of the complicated interactions existingbetween mechanical, thermal and diffusion phenomena. Ford lower to d1, an unrealistic increase in temperature appears,whereas for d superior to d2, temperature decreases (Fig. 15).In the same way, the effect of d on concentration depends onthe value of this coefficient (Fig. 16).

Physical AspectsAccording to the value of d, the effects observed arecompletely different. Couplings result in acceleration orslowing of mass transport and thermal transfer. Thiscoefficient associated with cTµ generates some complicatednon-linear effects, which are greatly dependent on gradients.Experimental studies will make it possible to validate or not

d d d d d= = = = =02

, 1 2

c

k

c

kT Tµ

µ

µ

µand

584


Figure 11

Profiles of expansion due to diffusion independent of kµ value.

Figure 12

Effect of expansion coefficients on radial displacement.

Linearloading step

Constant stepLinear


0 180 900 1800 2520 5400 9000 19 800 37 800 37 810 37 890 38 070 38 790 40 410 43 290 47 970 75 690 Time (s)

5 5.25 5.5 5.75 6Radius (mm)

0.E+00

4.E-02

3.E-02

2.E-02

1.E-02

Exp

ansi

on d

ue to

gas

diff

usio

n

t = 900 s

t = 5400 s

t = 37 800 s

t = 1800 s

t = 9000 s

t = 180 s

t = 2520 s

t = 19 800 s

5 5.25 5.5 5.75 6Radius (mm)

0.E+00

4.E-02

3.E-02

2.E-02

1.E-02

Exp

ansi

on d

ue to

gas

diff

usio

n

t = 38 070 s

t = 43 290 s

t = 37 890 s

t = 40 410 s

t = 75 690 s

t = 37 810 s

t = 38 790 s

t = 47 970 s

0 37 800 75 600Time (s)

-9.E-03

3.E-03

0.E+00

-3.E-03

-6.E-03Rad

ial d

ispl

acem

ent (

mm

)

r = 5.25 mm

0 37 800 75 600Time (s)

-8.E-04

8.E-04

0.E+00

Rad

ial d

ispl

acem

ent (

mm

)

r = 5.75 mm

1.E8 with thermal and diffusion expansions



1.E30 without expansion

1.E8 with thermal expansion




0

10

5 6Radius (mm)

5

Con

cent

ratio

n (M

Pa)

-0.2

0.0

5 6Radius (mm)

Rel

ativ

e sc

atte

r of

con

cent

ratio

n (%

)

-5.E-03

0.E+00

5 6Radius (mm)

Abs

olut

e di

ffere

nce

ofco

ncen

trat

ion

(MP

a)

20

280

5 6

5 6

Radius (mm)

150Te

mpe

ratu

re (

°C)

20

215

5 6Radius (mm)

85

150

Tem

pera

ture

(°C

)

t = 37 890 s

t = 40 410 s

t = 75 690 s

t = 38 070 s

t = 43 290 s

t = 37 810 s

t = 38 790 s

t = 47 970 s

t = 900 s

t = 5400 s

t = 37 800 s

t = 1800 s

t = 9000 s

t = 180 s

t = 2520 s

t = 19 800 s

0.0

0.8

-0.70

0.35

5 6Radius (mm) Radius (mm)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

0

5

10

5 6Radius (mm)

Con

cent

ratio

n (M

Pa)

Linearloading step

Constant stepLinear


0 180 900 1800 2520 5400 9000 19 800 37 800 37 810 37 890 38 070 38 790 40 410 43 290 47 970 75 690 Time (s)

kµ = 104 kg.s/m3 and cTµ = 10-2 kg/(m.s.K)

Figure 13

Effect of a small cTµ value: 10–2 kg/(m·s·K). Comparison with a case without cTµ.


Figure 14

Effect of a high cTµ value: 103 kg/(m·s·K). Comparison with a case without cTµ.

0

10

5 6Radius (mm)

5

Con

cent

ratio

n (M

Pa)

-100

100

5 6Radius (mm)

Rel

ativ

e sc

atte

r of

con

cent

ratio

n (%

)

0

8

5 6Radius (mm)

Abs

olut

e di

ffere

nce

ofco

ncen

trat

ion

(MP

a)

20

215

5 6Radius (mm)

5 6Radius (mm)

85

150

Tem

pera

ture

(°C

)

205 6

Radius (mm)

85

150

Tem

pera

ture

(°C

)

t = 900 s

t = 5400 s

t = 37 800 s

t = 1800 s

t = 9000 s

t = 180 s

t = 2520 s

t = 19 800 s

-42

0

-60

60

5 6radius (mm)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

0

5

10

5 6Radius (mm)

Con

cent

ratio

n (M

Pa)

Linearloading step

Constant stepLinear


0 180 900 1800 2520 5400 9000 19 800 37 800 37 810 37 850 37 880 37 890 Time (s)

kµ = 104 kg.s/m3 and cTµ = 103 kg/(m.s.K)

t = 37 850 s

t = 37 890 s

t = 37 810 s

t = 37 880 s


-45

15

5 6Radius (mm)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

0

60

5 6Radius (mm)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

-2

10

5 6Radius (mm)

5 6Radius (mm)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

05 6

Radius (mm)

8

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

-90

0

-30

90

5 6Radius (mm)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

Rel

ativ

e sc

atte

r of

tem

pera

ture

(%

)

10

320

0 75 600Time (s)

Tem

pera

ture

(°C

)

Linearloading step

Constant stepLinear


0 180 900 1800 2520 5400 9000 19 800 37 800 37 810 40 410 43 290 47 970 75 69038 070 38 79037 890 Time (s)

kµ = 104 kg.s/m3, cTµ = 1 kg/(m.s.K)

t = 37 890 s

t = 40 410 s

t = 59 490 s

t = 38 070 s

t = 43 290 s

t = 37 810 s

t = 38 790 s

t = 47 970 s t = 75 690 s

t = 900 s

t = 5400 s

t = 37 800 s

t = 1800 s

t = 9000 s

t = 180 s

t = 2520 s

t = 19 800 s

d = 3.E-4

d = 3.E-5

d = d1 = 1.E-4 d = d1 = 1.E-4

d = 3.E-5

r = 5.25 mm

5.E-5

3.E-4

3.E-5

1.E-4

0

7.E-510

165

0 75 600Time (s)

Tem

pera

ture

(°C

)

5.E-5

3.E-4

3.E-5

1.E-4

0

7.E-5

r = 5.75 mm

d = 3.E-4

Figure 15

Effects of cTµ and d on the thermal problem for kµ = 104 kg·s/m3, cTµ = 1 kg/(m·s·K) and d varying around d1 and d2. Comparison with thecase without cTµ and d.


Figure 16

Effect of cTµ and d on the mechanical and diffusion problems for kµ= 104 kg·s/m3, cTµ = 1 kg/(m·s·K) and d varying around d1 and d2.Comparison with the case without cTµ and d.

-0.55

0.05

5 6Radius (mm)

5 6Radius (mm)

Abs

olut

e di

ffere

nce

ofco

ncen

tatio

n (M

Pa)

-0.65 6

Radius (mm)

0

Abs

olut

e di

ffere

nce

ofco

ncen

trat

ion

(MP

a)

-0.8

0.2

-0.2

2.2

5 6Radius (mm)

Abs

olut

e di

ffere

nce

ofco

ncen

trat

ion

(MP

a)

Abs

olut

e di

ffere

nce

ofco

ncen

tatio

n (M

Pa)

-9.E-030 75 600

Time (s)

Rad

ial d

ispl

acem

ent (

mm

)

kµ = 104 kg.s/m3, cTµ = 1 kg/(m.s.K)

t = 37 890 s

t = 40 410 s

t = 59 490 s

t = 38 070 s

t = 43 290 s

t = 37 810 s

t = 38 790 s

t = 47 970 s t = 75 690 s

t = 900 s

t = 5400 s

t = 28 800 s

t = 1800 s

t = 9000 s

t = 180 s

t = 2520 s

t = 19 800 s t = 37 800 s

d = 3.E-4

d = 3.E-5 d = 3.E-5

r = 5.25 mm

r = 5.25 mm

00 75 600

Time (s)

Con

cent

ratio

n (M

Pa)

5.E-5

3.E-4

3.E-5

1.E-4

0

7.E-5

d = 3.E-4

Linearloading step

Constant stepLinear


0 180 900 1800 2520 5400 9000 19 800 37 800 37 810 40 410 43 290 47 970 75 69038 070 38 79037 890 Time (s)

8r = 5.75 mm

r = 5.75 mm

00 75 600

Time (s)

Con

cent

ratio

n (M

Pa)

2.5

3.E-03

-9.E-040 75 600

Time (s)

Rad

ial d

ispl

acem

ent (

mm

)

9.E-04

5.E-5 3.E-4

3.E-5 1.E-40 7.E-5

5.E-5

3.E-4

3.E-5

1.E-4

0

7.E-5

5.E-5

3.E-4

3.E-5

1.E-4

0

7.E-5


these first results. After solving of convergence problems, amore detailed parametric study on cTµ and d will lead tomore general conclusions.

CONCLUSION

The numerical tool developed in this paper makes it possibleto account for the direct couplings between mechanical,thermal and diffusion phenomena within an elasticframework. This exploratory and parametric study led tosome interesting results listed below.

The coefficient kµ, which corresponds to the effect ofdiffusion on thermal evolution, is an essential parameter. Itsstrong impact on the field of temperatures was underlined fora large set of values of kµ. This parameter governs the heatgeneration due to the gas diffusion within the polymer.Although this view of phenomena does probably notcorrespond completely to the mixture theory, the effects“naturally” present in the framework of porous media arereintroduced. Indeed, the flow of a viscous fluid through theskeleton induces an increase in temperature via a mechanicaleffect. With the aim of improving this modelling, it will benecessary to look into the signification of such a phenom-enon and to propose in the short term experimental tests toquantify the effect of kµ. Moreover, a simple modelling,including merely this coupling coefficient whose value islimited to a specific interval, describes de facto uncoupledbehaviour.

The second interesting result appears when introducingthe direct couplings with mechanics. They are represented bythe more classical coefficients of expansion, αT and αc,linked to the variations of temperature and concentration,respectively. In such a context, it becomes impossible toeliminate the effect of kµ; in other words the three couplingcoefficients, kµ, αT and αc, define the basic elasticmodelling. Moreover, during the respective simulationsof tubular structure under plane-strain conditions, strainphenomena were too slow to induce heat generation. It willbe thus essential to test some different boundary-valueproblems in further developments.

The two further coefficients, cTµ and d, produce muchmore subtle effects, linked to the gradients of temperature,concentration as well as volumetric swelling strain. Theeffect of these parameters could probably be explained moreprecisely in the presence of strong gradients; depending ongeometry, loading and boundary conditions inducing suchconfigurations should be considered.

It is also important to underline that, with simpleconstitutive laws, i.e. developed in a linear framework(quadratic forms of state and dissipation potentials), strongnonlinearities appear due to coupling effects. Even if thiswork made it possible to approach some specific effects of

coupling, further simulations will be necessary to captureother features of the relative phenomena.

The formulation, implemented in ABAQUS™, containsall direct couplings in the restricted mechanical framework ofelasticity. As polymers exhibit a viscous behaviour for smallloadings and deformations, a further extension to visco-elasticity is to be done. Within such enlarged framework, thecontribution of each direct coupling parameter will be moreprecisely quantified and the genuine polymer characterizationeasier. Indeed, expressions proposed to describe the intrinsicdependencies of material characteristics (i.e. indirect cou-plings) are much complicated and relative experimental datanot easy to exploit making necessary numerical approachesand relative parametric studies as the present one. On thisbasis, it is hopefully expected to propose a new protocol ofcharacterization. A practical discernment between direct andindirect couplings effects is also to be brought forward.

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Final manuscript received in August 2003

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a modelling of the coupled thermodiffuso-elastic linear

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