mathematics and mechanics of solids diffusion of...

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Mathematics and Mechanics of Solids

2015, Vol. 20(2) 204–227

� The Author(s) 2014

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DOI: 10.1177/1081286514544852

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Diffusion of chemically reacting fluidsthrough nonlinear elastic solids: mixturemodel and stabilized methods

R HallMaterials and Manufacturing Directorate, Air Force Research Laboratory, OH, USA

H Gajendran and A MasudDepartment of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Received 8 November 2013; accepted 12 March 2014

AbstractThis paper presents a stabilized mixed finite element method for advection-diffusion-reaction phenomena that involve ananisotropic viscous fluid diffusing and chemically reacting with an anisotropic elastic solid. The reactive fluid–solid mixturetheory of Hall and Rajagopal (Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body withinthe context of mixture theory. Math Mech Solid 2012; 17: 131–164) is employed wherein energy and entropy productionrelations are captured via an equation describing the Lagrange multiplier that results from imposing the constraint ofmaximum rate of entropy production. The primary partial differential equations are thus reduced to the balance of massand balance of linear momentum equations for the fluid and the solid, together with an equation for the Lagrange multi-plier. Present implementation considers a simplification of the full system of governing equations in the context of iso-thermal problems, although anisothermal studies are being investigated. The method is applied to problems involvingFickian diffusion, oxidation of PMR-15 polyimide resin, and slurry infiltration, within a one-dimensional finite element con-text. Results of the oxidation modeling of Tandon et al. (Modeling of oxidative development in PMR-15 resin. PolymDegrad Stab 2006; 91: 1861–1869) are recovered by employing the reaction kinetics model and properties assumedthere; the only additional assumed properties are two constants describing coupled chemomechanical and purely chemi-cal dissipation, and standard values for viscosity of air and PMR-15 stiffness properties. The present model provides theindividual constituent kinematic and kinetic behaviors, thus adding rich detail to the interpretation of the process in com-parison to the original treatment. The last problem considered is slurry infiltration that demonstrates the applicability ofthe model to account for the imposed mass deposition process and consequent effects on the kinematic and kineticbehaviors of the constituents.

KeywordsMixture theory, oxidation, slurry infiltration, stabilized method, variational multiscale method, PMR-15 resin

Dedicated to KR Rajagopal.

Corresponding author:

A Masud, Professor of Mechanics and Structures. Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205

North Mathews Ave., Urbana, IL 61801-2352, USA.

Email: [email protected]

1. Introduction

In this paper, a stabilized mixed finite element method is presented for the diffusion of a chemicallyreacting fluid through a nonlinear elastic solid using a mixture theory based model. For a detailed intro-duction to mixture theory, interested readers are referred to comprehensive review articles by Atkin andCraine [1], Green and Naghdi [2,3], and the book by Rajagopal and Tao [4]. Mixture theory ideas havebeen used to model various phenomena, such as classical viscoelasticity [5], swelling of polymers [6],thermo-oxidative degradation of polymer composites [7,8], and growth of biological materials [9] andcrystallization of polymers [10]. Malek and Rajagopal [11] proposed that processes for fluid mixtures aregoverned by the maximization of the rate of dissipation constraint. Karra [7] developed a mixture theorymodel and its constitutive relations based on this constraint for diffusion of a fluid through a viscoelasticsolid. Karra and Rajagopal [8] also developed a mixture theory model for degradation of polyimides dueto oxidation. A limitation of their model is that it cannot predict the oxidation layer thickness growth.Hall and Rajagopal [12] proposed a mixture theory model for diffusion of chemically reacting fluidthrough an anisotropic solid. Their model is based on the maximization of the rate of entropy produc-tion constraint, considering anisotropic effective reaction rates and the limits of diffusion-dominated(diffusion of the reactants is far more rapid than the reaction) and reaction-dominated (the reaction isfar more rapid than the diffusion of the reactants) processes. This model in general can be applied to avariety of processes that involve directionality of flow, directionality of the reaction process, and that ofthe solid medium. Such processes arise in, as examples, the curing of composites using vacuum-assistedresin transfer molding (VARTM), in the prediction of oxidation layer growth in composites, and inslurry infiltration (SI) in the manufacturing of composites.

The mixture theory model combines the composite constituent behaviors in an effective mediumsense, reducing the computational cost of modeling chemically reacting multi-constituent mixtures,while retaining information involving the kinematic and kinetic responses of the individual constituents.The effective medium and individual constituent behaviors are each constrained to mutually satisfy thebalance principles of mechanics.

In this work we employ Variational Multiscale (VMS) framework [13–16] to develop a stabilizedmixed finite element formulation involving the balance of mass equation for the fluid that is written inan Arbitrary Lagrangian Eulerian (ALE) form. The underlying idea in VMS is an additive decomposi-tion of the solution field into coarse- and fine-scale components, which in the present context leads to amultiscale decomposition of the fluid density field that results in two coupled nonlinear problemstermed as the coarse-scale and the fine-scale sub-problems. The space of coarse scales is identified withthe standard finite element functional spaces, while the space of fine scales can contain various finite-dimensional approximations that are subject to the condition that the spaces are linearly independent.The fine-scale solution is extracted from the nonlinear fine-scale sub-problem, and it is then variation-ally embedded in the coarse-scale equation, leading to a formulation that is expressed entirely in termsof the coarse scales. Consequently, the resulting stabilized form does not have an explicit appearance ofthe fine-scale density field of the fluid. Rather, the effects of fine scales are represented via the additionalresidual-based terms that in fact add to the stability of the numerical method.

One of the applications of interest in the present work is the oxidation of polymer matrix composites(PMCs). Tandon et al. [17] conducted experiments to study oxidation processes in a high-temperaturepolyimide resin used in aerospace composites, and developed an oxidation reaction rate model that con-forms to the observed experimental data. In this work, we implement this oxidation model in the con-text of mixture theory. Schoeppner et al. [18] and Varghese et al. [19] and Varghese and Whitcomb [20]developed finite element algorithms for the diffusion-reaction equation to model the oxidation in PMR-15 resin and PMCs. In their work, the fibers and matrix were modeled in a discrete sense and thus theiralgorithm was computationally intensive. Varghese et al. [19] proposed an adaptive mesh strategy anddecoupled subdomain strategy to reduce the computational cost of their algorithm. Their adaptive meshstrategy requires a prior knowledge of oxidation layer growth to constrain the unoxidized region, thusreducing the number of unknowns in the problem.

The outline of the paper is as follows. In Section 2, we present the governing equations and the con-stitutive relations derived from the mixture theory for a chemically reacting fluid diffusing through anonlinear elastic solid in a general three-dimensional context. The modeling assumptions and the one-dimensional form of the general mixture theory are presented in Section 3. In Section 4, we present the

Hall et al. 205

weak form of the mixture theory governing equations and develop the VMS-based stabilized method.Section 5 presents the finite element results of the mixture theory for Fick’s diffusion problem, in thecontext of matching an analytical solution for demonstration of accuracy and stability of the numericalapproach; oxidation of PMR-15 resin; and SI in PMCs. Conclusions are drawn in Section 6.

2. General mixture theory

In this section, we first present the mixture theory-based model for diffusion of an anisotropic nonlinearviscoelastic fluid through an anisotropic elastic solid with mutual chemical reaction, as proposed by Halland Rajagopal [12]. A basic assumption in the mixture theory is that the constituents of the mixture co-occupy the domain and as the mixture deforms, these co-existing continua deform with respect to eachother. A set of appropriate constitutive relations that are based on the constraint of maximum rate ofentropy production are also presented in [12]. In the present work, we consider the constitutive relationsassociated with unconstrained constituent volumes. Detailed derivation is available in [12].

The equations of mass and linear momentum balance for the diffusion of a chemically reacting fluidthrough a finitely deforming thermoelastic solid are given as follows [12]:

Balance of mass :Dara

dt+ radivva =

∂ra

∂t+div(rava) = ma ð1Þ

Balance of linear momentum: ra Dava

dt= div(Ta)T + rab+ Ia ð2Þ

where ra is the mass concentration and ma is the rate of mass transferred by chemical reaction, to constituent a,per unit mixture volume; va is the velocity of constituent a and Ta is its partial Cauchy stress, while Ia and b arethe interactive force per unit mixture volume on constituent a and the body force per unit mass.

The balance of energy and assumption of maximized rate of entropy production, together withNewton’s third law, lead to the following relations for the partial stresses on the solid and fluid, T s andT f ; the interactive force I f on the fluid, the constituent entropy ha, and the rate of fluid mass conver-sion, m f , all per unit mixture volume; and the heat flux q, per unit mixture area:

I f = g f rs

rrr f � gs r f

rrrs �r rsr f

r(c f � cs)

� �

�(ru)rsr f

r(hf � hs)� m f (v f � vs)� mAv(v f � vs)

ð3Þ

T s = rF s(∂c

∂F s )T � rs(gs +r f

r(c f � cs))I ð4Þ

T f = � r f (g f +rs

r(cs � c f ))I+ mAL �D f ð5Þ

ha = � ∂ca

∂u� m

rca

u_u ð6Þ

q

u= � mlru +

rsr f

r(hf � hs)(v f � vs) ð7Þ

m f =1

mcm�(g f � gs)� 1

2(v f � vs) � (v f � vs)

� �ð8Þ

where the chemical potential ga of constituent a is defined through

ga[r∂c

∂rað9Þ

206 Mathematics and Mechanics of Solids 20(2)

r, c, and u are the mixture density, mixture Helmholtz energy, and temperature, respectively, while ca

are the constituent Helmholtz energies; material parameters cau and cm are respectively associated with

the constituent entropies, and with mass transfer, while l is the mixture thermal conductivity tensor; F s

is the solid deformation gradient, Av and AL are drag and viscosity coefficient tensors, andD f is the fluidrate of deformation tensor.

The rate of mass transfer to the fluid, m f , is determined in coordination with the orientation averageof the rate of reaction tensor _G. Because of the presence of only two constituents, the mass balance pro-vides that the rate of mass converted to the solid is the one lost from the fluid:

ms = �m f ð10Þ

In the diffusion-dominated approximation (diffusion of the reactants is far more rapid than the reac-tion), the operator _Pfn,Xs, tg provides the directional solid mass conversion rate in the direction 2n,per unit mixture volume, such that:

ms =1

4p

ð4p

a = 0

_Pfn½a�,Xs, tgda ð11Þ

where n is the outward unit normal, Xs is the reference coordinate of the solid, a is the solid angle, and asecond-order representation is assumed for the operator _Pfn,X s, tg:

_Pfn,Xs, tg’n � _G ½Xs, t� n ð12Þ

with the tensor

G ½Xs, t�=ðt0

_G ½Xs,�t� d�t ð13Þ

thus providing an anisotropic measure of the extent of reaction of the solid.Employing in the present work the Lagrangian solid strain measure Es and referring G to material

coordinates, the Lagrange multiplier arising from the constraint of maximized rate of entropy produc-tion is given by, in the general case, cf. [12]:

m =1

2+

1

4_G0

IJ G0KLG0

MN EsOPK0

IJKLMNOP �ru � rsr f

r(hf � hs)(v f � vs)� m f (v f � vs) � (v f � vs)

� �D f � AL �D f + cu

_u2 + (v f � vs) � Av(v f � vs) + _G0 � A0G � _G0 +ru � lru + cm(m f )2

+1

2_G0

ABG0CDG0

EFEsGHK0

ABCDEFGH

( ) ð14Þ

where K0IJKLMNOP is a tensor that couples the mechanical and chemically influenced attributes of the

model, in a way that is compatible with the results of the maximization of the rate of entropy produc-tion as described in [12]. Because ha and m f depend on m, Equation (14) is a cubic equation in m. Toobtain a single-valued relation for m, the following approximations are made:

1. We assume that the attributes of the Helmholtz free energy functions of the constituents and themixture can be represented in terms of suitably condensed forms, cs = c f = c, hs = hf = h;

2. Slow diffusion permits neglect of the squared relative kinetic energy terms ((v f � vs) � (v f � vs))2,which are assumed also negligible relative to the drag force;

3. We assume that the reaction is near enough to equilibrium to neglect the squared difference in thechemical potentials of the constituents, and the product of the chemical potential difference withthe relative kinetic energy.

The Lagrange multiplier is thus reduced to the following single-valued function:

Hall et al. 207

m =1

2+

14

_G0IJ G0

KLG0MN Es

OPK0IJKLMNOP

D f � AL �D f + cu_u2 + (v f � vs) � Av(v f � vs) +

_G0 � A0G � _G0 +ru � lru + 12

_G0ABG0

CDG0EFEs

GH K0ABCDEFGH

� � ð15Þ

It can be noted that the tensor K0 will have mostly zero-valued components. If reaction processes suchas oxidation are considered, in which the reaction is several times faster in the fiber direction than thetransverse directions, thus promoting a unidirectional reaction assumption, and assuming transverselyisotropic coupling to the strains, the term involving K0 reduces to the following expression, involvingfour independent constants:

_G0IJ G0

KLG0MN Es

OPK0IJKLMNOP = _G0

11(G011)2 K0

1 Es11 + K0

2 (Es22 + Es

33) + K03 (Es

12 + Es31) + K0

4 Es23

� �ð16Þ

In the present work, the influence of the energy and entropy production relations is retained through thepresence of the Lagrange multiplier, which is obtained via invoking the constraint of maximized rate ofentropy production. The equations explicitly retained are the constituent momentum balances and themass balance equation, which can be considered most strongly enforced. In accordance with the presentstudy being isothermal, the traditional heat capacity measures of the constituents are lost through theassumption above that the constituent entropy functions can be replaced by an overall entropy function.In general for anisothermal processes, the Helmholtz and entropy functions of each constituent wouldbe retained. It is interesting to note, however, that the present system of equations incorporates the rateof temperature in combination with a non-traditional overall material property cu (the density averageof the ca

u properties), which may provide a simplified approach to accounting for a class of homogenizedanisothermal effects. The present paper, however, considers only isothermal conditions.

3. One-dimensional mixture theory

Consider a one-dimensional mixture domain O of length L with boundary ∂O= x x 2 0, Lf gjf g. The gov-erning equations for the one-dimensional case under isothermal conditions are as follows:

Balance ofmass :∂ra

∂t+∂ra

∂xva

1 + ra ∂va1

∂x� ma = 0 ð17Þ

Balance of linearmomentum :∂Ta

11

∂x+ rab1 + Ia

1 � ra Dva1

Dt= 0 ð18Þ

The corresponding stresses and interactive force on the constituents can be written as follows:

Ts11 = rF11

∂c

∂F11

� rs(r∂c

∂rs+

r f

r(c f � cs)) ð19Þ

Tf11 = � r f (r

∂c

∂r f+

rs

r(cs � c f )) + mAL ∂v

f1

∂xð20Þ

If1 =

∂c

∂r frs ∂r f

∂x� ∂c

∂rsr f ∂rs

∂x� ∂

∂x

rsr f

r(c f � cs)

� ��

∂u

∂x

rsr f

r(hf � hs)� m f (v f

1 � vs1)� mAv(v f

1 � vs1)

ð21Þ

We consider the following Helmholtz free energy function that corresponds to the one-dimensionalrepresentation of a transversely isotropic thermoelastic solid permeated by a chemically reactingNewtonian fluid:


c = As + (Bs + cs)(u� us)� cs1

2(u� us)2 � cs

2u lnu

us

� +

1

rT

�Rur f + kf2r f

n o

+rs

r

1

rsT

1

2ls + ms

T + as + 2(msL � ms

T ) +1

2bs

� �(Es

11)2 + L

ð22Þ

L =

ðm

2(G0

11)2 �K1Es11 + 2 _G0

11�A0G

n odG0

11 ð23Þ

where L describes the coupling between the solid strain and the extent of reaction, consistent with thedevelopments of [12]; ls, as,ms

L, msT , bs are the transversely isotropic material constants, which in one

dimension reduce to the elastic moduli of the solid. rsT , rT are the true solid density and the true mix-

ture density, respectively. �R is the ratio of the universal gas constant to the molecular weight of the fluid.K0

1 = � r �K1 and A0G = � r�A0G are defined for convenient manipulations involving L.

Remark: For the case of the slurry deposition process that is presented in Section 5.3, G011 represents the extent of material

deposition. For this case, the term L provides coupling between the solid strain and the extent of deposition of the sus-pended particles. We assume that this deposition function G0

11 is in fact a function of the volume fraction of particles, whichis considered a process parameter.

The one-dimensional representation of the Lagrange multiplier m is given as

m =1

2�

14

_G011(G0

11)2r �K1Es11

AL(∂v

f

1

∂x)2 + cu

_u2 +Av(v f1 � vs

1)2 � r �A0G( _G011)2 + I11( ∂u

∂x)2 � 1

2_G0

11(G011)2r �K1Es

11

ð24Þ

Also, from mass balance law and Newton’s third law we see that the solid and the fluid interactive forceshave the following relationship:

I s1 + msvs

1 = � (I f1 + m f vs

1) ð25Þ

3.1 Modeling assumptions and methodology

In mixture theory where both solid and fluid co-occupy the domain and fluid moves relative to thedeforming solid, it is natural to write the fluid balance laws in an ALE framework [21–23]. For the classof problems considered in this work, the inertial effects on the solid are assumed to be negligible. Basedon these modeling assumptions, the balance laws Equations (17) and (18) can be rewritten as follows:

∂rs

∂t+∂rs

∂xvs

1 + rs ∂vs1

∂x� ms = 0 ð26Þ

∂Ts11

∂x+ rsb1 + I s

1 = 0 ð27Þ

∂r f

∂t

Y

+∂r f

∂x(v f

1 � vm1 ) + r f ∂v

f1

∂x� m f = 0 ð28Þ

∂Tf11

∂x+ r f b1 + I

f1 � r f ∂v

f1

∂t

Y

�r f (v f1 � vm

1 )∂v

f1

∂x= 0 ð29Þ

where∂( � )∂t

Y

represents the time derivative in the ALE frame [22,23] and vm1 is the fluid mesh velocity.

It is important to note that as the solid domain deforms, the Lagrangian mesh that is tied to materialpoints deforms together with it. Consequently, the mesh velocity vm

1 is set equal to vs1, where vs

1 is thevelocity of the solid domain. Accordingly, the constitutive relations can be rewritten as

Hall et al. 209

Ts11 = rF11

∂c

∂F11

� rsr∂c

∂rsð30Þ

Tf11 = � r f r

∂c

∂r f+ mAL ∂vf

∂xð31Þ

If1 = � m f (v f

1 � vs1)� mAv(v f

1 � vs1) ð32Þ

m f = _G011 ð33Þ

Remark: In [12] an expression for the rate of mass conversion for fluid m f is derived via maximization of the rate of dissipa-tion constraint. However, in the present work we prescribe an oxidation rate given in [17] that is developed based on physi-cal measurements. Likewise, in the SI model we prescribe a rate of particle deposition as is given in [24]. Because of thesepostulated rates, the physics involved in the consistent derivation of mass conversion given in [12] is circumvented.

4. Weak form and development of the stabilized method

The initial conditions for the density and velocity fields of the two constituents and the displacementfield of the solid are

ra(x, 0) = ra0 ; va

1 (x, 0) = va0 ; us

1(x, 0) = us0 8x 2 O ð34Þ

The boundary ∂O admits decomposition into ∂Og and ∂Oh, where ∂Og \ ∂Oh = f, and we denote the unitnormal to the boundary ∂O by n1. The boundary conditions for the problem are

r f = rf0 on ∂Or f

g 3 0, T� ½v

f1 = v

f0 on ∂O f

g 3 0, T� ½us

1 = us0 on ∂Os

g 3 0, T� ½T

f11n1 = t

f0 on ∂Of

h 3 0, T� ½Ts

11n1 = ts0 on ∂Os

h 3 0, T� ½

ð35Þ

where rf0, v

f0 are the prescribed fluid density and velocity, and us

0 is the prescribed solid displacement. tf0

and ts0 represent the prescribed fluid and solid boundary tractions, respectively.

Let ga and wa1 denote the weighting functions for the balance of mass and linear momentum for the

corresponding constituent, respectively. The appropriate spaces for these weighting functions are

V = gajga 2 H1(O), ga = 0 on ∂Ora

g

n oð36Þ

Q= wa1

wa1 2 H1(O),wa

1 = 0 on ∂Oag

n oð37Þ

The corresponding trial solution spaces for the fluid and solid density, fluid velocity, and solid displace-ment are

Sra

= ra(�, t)jra(�, t) 2 H1(O), ra(�, t) = ra0 on ∂Ora

g 3 0,T� ½n o

ð38Þ

Sf = vf1 (�, t)

v f1 (�, t) 2 H1(O), v

f1 (�, t) = v

f0on ∂O f

g 3 0,T� ½n o

ð39Þ

Ss = us1(�, t)

us1(�, t) 2 H1(O), us

1(�, t) = us0 on ∂Os

g 3 0,T� ½n o

ð40Þ


The weak form of governing equations for the solid–fluid system can be stated as follows. For all consti-tuents a 2 s, ff g, 8t 2 (0, T ), 8ga 2 V, and 8wa

1 2 Q, solve ra 2 Sra

, vf1 2 Sf , and us

1 2 Ss such that thefollowing system holds.

Weak form of equations for the fluid:

g f ,∂r f

∂t

Y

� + g f , (v f

1 � vs1)∂r f

∂x

� + g f , r f ∂v

f1

∂x

!� (g f ,m f ) = 0 ð41Þ

∂wf1

∂x,T

f11

!� (wf

1, rf b1)� (w f

1 , If

1 ) + wf1 , r f ∂v

f1

∂t

Y

!+

wf1, r

f (v f1 � vs

1)∂v

f1

∂x

!� (wf

1,Tf11n1)

∂Of

h

= 0

ð42Þ

Weak form of equations for the solid:

gs,∂rs

∂t

� + gs, rs ∂vs

1

∂x

� + gs, vs

1

∂rs

∂x

� � (gs,ms) = 0 ð43Þ

∂ws1

∂x,Ts

11

� � (ws

1, rsb1)� (ws

1, Is1)� (ws

1, Ts11n1)∂Os

h= 0 ð44Þ

where �, �ð Þ=RO�ð ÞdO is the L2 Oð Þ inner product.

4.1 Fluid sub-system: residual-based stabilization

Our objective is to model the diffusion of a chemically reacting fluid through a nonlinear elastic solid, aphenomenon that is observed in the process modeling of composites, oxidation of resin/composites, andSI in porous media, to name a few. In the modeling of these processes, fluid mass concentration is invari-ably specified at the inlet boundary. Since the strong form of mass balance of fluid given in Equation(17) is a first-order hyperbolic equation, any specified mass concentration boundary condition at theinlet that is different from the initial condition results in a discontinuous fluid concentration field. Thisdiscontinuity introduces spurious oscillations in the computed solution right at the beginning of the non-linear iterative process that can lead to non-convergent and therefore non-physical solutions.

To address this issue, we consider the weak form of the balance of mass equation for the fluid that iswritten in an ALE form. We employ VMS ideas [13–16] and develop a stabilized weak form forEquation (41). Underlying idea of VMS is an additive decomposition of the solution field into coarse-and fine-scale components as given below:

r f = r f + ~r f ð45Þ

g f = g f + ~g f ð46Þ

where rf , ~rf represents the coarse-scale and fine-scale components of the density field and g f , ~g f repre-sents the coarse-scale and fine-scale counterpart of the weighting function, respectively. Various scaleseparations of rf are possible in Equation (45). However, they are subject to the restriction imposed bythe stability of the formulation that requires the spaces for the coarse-scale and fine-scale functions tobe linearly independent. In the development presented here, the spaces of coarse-scale weighting func-tions are identified with the standard finite element spaces, while the fine-scale weighting functions cancontain various finite-dimensional approximations, for example bubble functions or p-refinements orhigher order Non-Uniform Rational B-Spline (NURBS) functions.

Substituting Equations (45) and (46) into Equation (41) and employing the linearity of the weightingfunction slot in Equation (41), we obtain the coarse-scale problem and the fine-scale problem as given inEquations (47) and (48), respectively:

Hall et al. 211

Wr f

= g f ,∂(r f + ~r f )

∂t

Y

� + g f ,

∂(r f + ~r f )v f1

∂x

!� g f , vs

1

∂(r f + ~r f )

∂x

� � (g f ,m f ) = 0 ð47Þ

~Wr f

= ~g f ,∂(r f + ~r f )

∂t

Y

� + ~g f ,

∂(r f + ~r f )v f1

∂x

!� ~g f , vs

1

∂(r f + ~r f )

∂x

� � (~g f ,m f ) = 0 ð48Þ

It is important to note that both systems are nonlinear, and are also fully coupled in terms of the scales.The key idea at this point is to solve the fine-scale problem Equation (48) locally, using analytical meth-ods or numerical methods, and extract the fine-scale component, ~rf . This can then be substituted intothe corresponding coarse-scale problem given in Equation (47), thereby eliminating the fine scales, yetmodeling their effects.

4.1.1 Solution of the fine-scale problem. We segregate the terms into coarse-scale and fine-scale terms and groupall the terms containing a coarse-scale density field:

~Wr f

= ~g f ,∂~r f

∂t

Y

� + ~g f ,

∂~r f vf1

∂x

!� ~g f , vs

1

∂~r f

∂x

� + (~g f , R) = 0 ð49Þ

where R is the residual of the Euler–Lagrange equations of the coarse scales over element interiors and isgiven as

R =∂r f

∂t

Y

+∂r f v

f1

∂x� vs

1

∂r f

∂x� m f (r f ) ð50Þ

In obtaining the above form of the fine-scale problem, we have assumed that the fluid mass conversionrate is a function of the coarse-scale fluid density field only, m f (r f , ~r f )’m f (r f ). To reduce the complex-ity of the fine-scale problem and also to reduce the computational cost for evaluating the fine-scale solu-tion field, we assume that the fine-scale field vanishes at the element boundaries:

~g f = 0, ~rf = 0 on Ge ð51Þ

Remark: The assumption that fine scales vanish at the inter-element boundaries helps in keeping the presentation of theideas simple and concise. Relaxing this assumption in fact leads to a more general framework. This, however, requiresLagrange multipliers to enforce the continuity of the fine-scale fields across inter-element boundaries. It is important tonote that Lagrange multipliers can be accommodated in the present hierarchical framework as well.

Using the backward Euler time integration scheme and assuming that the fine-scale fluid density field atthe beginning of a time step is zero, ~rf

n = 0, we can obtain the time discretized form of Equation (49) asgiven below:

~Wr f

= ~g f ,~r f

Dt

Y

� + ~g f ,

∂~r f vf1

∂x

!� ~g f , vs

1

∂~r f

∂x

� + (~g f , R) = 0 ð52Þ

The fine-scale fields are represented by bubble functions within each element and are given as

~gfn + 1 = be

2�gfn + 1, ~rf

n + 1 = be1�rf

n + 1 ð53Þ

where be1, be

2 are bubble functions and �gfn + 1, �r

fn + 1 are the coefficients associated with the fine-scale fields

over the element; examples are shown schematically in Figure 1.Substituting Equation (53) into Equation (52), and following along Masud and Khurram [15] and

Calderer and Masud [23], we can obtain the fine-scale density field via solution of Equation (52) asfollows:


~r f = tR ð54Þ

where �R is the mean value of the residual of the Euler–Lagrange equations of the coarse scales for

Equation (47). The stabilization parameter, t, is given as

t =�be

1(be2, 1)Oe

(be2,

1Dt

be1)Oe

+ (be2,

∂vf

1

∂xbe

1)Oe+ (be

2, vf1

∂be1

∂x)Oe� (be

2, vs1

∂be1

∂x)Oe

ð55Þ

We now substitute the fine-scale solution given in Equation (54) into the coarse-scale problem,Equation (47):

Wr f

= g f ,∂r f

∂t

Y

� + g f ,

∂r f vf1

∂x

!� g f , vs

1

∂r f

∂x

� � (g f ,m f )

+ (g f ,1

Dt+∂vs

1

∂x

� t�R)� ∂g f

∂x, v

f1 � vs

1

� �t�R

� = 0

ð56Þ

Equation (56) represents the modified coarse-scale problem with the fine-scale effects embedded impli-citly via the coarse-scale residual terms. The first four integral terms in Equation (56) correspond to thestandard Galerkin method for the balance of mass for the fluid. The last two terms in Equation (56) haveappeared because of the fine-scale density field. It is important to note that the fine-scale density doesnot explicitly appear in Equation (56); rather, the fine-scale effects are implicitly reflected in this formvia the modeling terms.

Equations (56), (42), (43), and (44) are linearized and solved simultaneously for the density and velo-city fields of the fluid, and density and displacement fields of the solid using Newton–Raphson solutionprocedure. This coupled system of equations is discretized-in-time using the backward Euler scheme,while linear and quadratic Lagrange elements with equal-order fields are employed in the spatial dimen-sion. The resulting stiffness matrix for the full system is non-symmetric.

5. Numerical results

We present three test cases that investigate the stability and accuracy of the numerical method developedfor the mixture theory model described in Section 4. In Section 5.1, we solve a reduced mixture model thatis equivalent to the Fick’s diffusion problem. A system comprising a first-order hyperbolic equation and analgebraic equation is solved and the results are compared with the exact solution. Section 5.2 presents theoxidation problem of PMR-15 resin wherein a full system of mixture theory equations is solved and theresults are compared with the experimental and numerical results reported by Tandon et al. [17]. Section5.3 models the SI process that is involved in the manufacturing of composites, and a parametric study ofthe reduction in the porosity of the solid as a function of slurry particle fraction and initial solid porosity ispresented. The present developments have been carried out in the context of the one-dimensional finite ele-ment method, and extension to the three-dimensional case will be pursued in a subsequent paper.

5.1 Fick’s diffusion problem

In this section we employ a reduced mixture model to solve Fick’s diffusion problem. The transientFick’s diffusion equation can be derived from the mixture theory balance laws, Equations (17) and (18),

Figure 1. Schematic representation of quadratic and linear bubbles.

Hall et al. 213

based on the following simplifications: (a) solid is assumed to be rigid; (b) fluid is assumed ideal; (c)fluid inertial effects are neglected; and (d) fluid is assumed non-reactive. The constitutive relations foran ideal fluid and the interactive force between the fluid and rigid solid can be given as [25]

Tf11 = � r f �Ru ð57Þ

If1 = � Avr f v

f1 ð58Þ

where Av is the drag coefficient. The governing Equations (17) and (18) can be reduced to the followingsystem of equations:

∂r f

∂t+∂(r f v

f1 )

∂x= 0 ð59Þ

� ∂rf

∂x�Ru� Avrf v

f1 = 0 ð60Þ

Since the coupled system of Equations (59) and (60) serves as a reduced-order model for the mixturetheory, we solve this first-order system to investigate the underlying characteristics of the mixture modelwherein the conservation of mass equation for the fluid is hyperbolic. The diffusivity of the solid can bewritten in terms of the drag coefficient of the solid as

D =�Ru

Avð61Þ

The derivation of Equation (61) is provided in the Appendix.

Remark: Solving for fluid velocity from Equation (60) and substituting back into Equation (59), one can obtain Fick’s dif-fusion equation. Since our full mixture model results in a first-order system, in this work we have opted to solve the reducedsystem also in its first-order form to help serve as a test case for evaluating our numerical method.

The unknown fields in this problem are the fluid concentration and fluid velocity that are solved withzero initial conditions. The one-dimensional domain of length 0.001m is exposed to air at the left endof the domain where fluid concentration is assigned a value of 22.8863E-3 kg/m3 and fluid velocity isconstrained to be zero at the right end of the domain. The gas constant �R and drag coefficient Av areassigned values of 286.987 J/kg-K and 1.63E17 s21, respectively. We employ the backward Euler schemefor time integration and run the problem for a total time of 30,000 seconds. A variable time step incre-ment is used: the time step employed during the first second is Dt = 1E-4, and it is increased to Dt = 0:1for the remaining steps.

It should be noted that Equation (59) is a first-order hyperbolic equation for fluid concentration. Fora non-zero fluid concentration boundary condition applied at the inflow, the standard Galerkin finiteelement method results in oscillations around the steep front thereby causing numerical instability. Weemploy the VMS method as described in Section 4 to stabilize the formulation, and provide a compari-son between the stabilized numerical result with the exact solution. Figures 2(a) and (b) show the per-formance of the new method for h-refinement wherein we have used linear Lagrange interpolationfunctions. These plots show the spatial profiles of the fluid concentration and velocity fields at 1000,10,000, and 30,000 seconds. It can be seen that as the number of elements is increased, the computedsolution converges to the exact solution, which is a numerical validation of the consistency of the for-mulation. Likewise, Figure 3 shows the convergence of the fluid density field for quadratic elements.Figure 4 shows the L2 norm of the error in the fluid density field for linear and quadratic VMS ele-ments. Convergence rates of 1.54 for linear and 1.88 for quadratic VMS elements are obtained for thenonlinear first-order problem. Figures 5(a) and (b) show that numerical results compare well with theexact solution at 1000, 10,000, and 30,000 seconds, wherein the domain is discretized with 400 elements.

In Figures 6(a) and (b), we show the spatial distributions of fluid density and fluid velocity for threedifferent values of the drag coefficient for a domain of length 1m. It can be seen that for a lower drag


Figure 2. Mesh refinement study at various time levels. (a) Fluid density: linear Lagrange h-refinement. (b) Fluid velocity: linearLagrange h-refinement.

Figure 3. Mesh refinement study using quadratic Lagrange elements.

Figure 4. Convergence plot of the L2 norm of fluid density.

Hall et al. 215

coefficient that corresponds to higher diffusivity, fluid propagates further down in the porous solid ascompared to the cases of higher drag coefficients.

5.2 Oxidation of PMR-15 resin

Thermo-oxidative aging of PMCs in high-temperature applications influences the life and performanceof these materials. In this section, we present numerical results for the oxidation behavior of polyimidePMR-15 resin based on the oxidation reaction model developed in the work of Tandon et al. [17]. Forthe sake of completeness, we provide a brief description of the oxidation process in polymer. However,for a detailed description of the oxidation process and the reaction kinetics model, refer to [17,18]. Theoxidation front in polymer materials advances through a combination of diffusion and reaction mechan-ism. The exposed surface reacts with the diffusing air that depletes the amount of polymer available inthat region. Once this region is fully oxidized, it acts as a medium through which air/oxygen diffuses andan active oxidation zone is formed ahead of the fully oxidized zone. Thus, at any instant of time, the oxi-dation process in polymers comprises a fully oxidized zone, an active oxidation zone, and a neat resinzone, as shown in Figure 7.

Figure 5. Comparison between exact and finite element solution. (a) Fluid density along the domain. (b) Fluid velocity along thedomain.

Figure 6. Variation of (a) fluid concentration and (b) fluid velocity for three different drag coefficients.


The oxidation reaction rate implemented in this work is given in [17] as

_G011 = (

f� fox

1� fox

)R(r f ) ð62Þ

R(r f ) =�R02j(1� 0:5j) f.fox

0 f�fox

�ð63Þ

j =�br f

1 + �br fð64Þ

where _G011 is the rate of reaction, f is the oxidation state variable that indicates the availability of poly-

mer for oxidation, fox specifies the fully oxidized state of the material (Zone I), R0 is the saturated rateof reaction, and �b is the inverse of the saturation air/oxygen concentration. The evolution equation forthe oxidation state variable f is given as

df

dt= am f ð65Þ

where a is the constant of proportionality. f varies in the active oxidation region while it assumes avalue of fox in the fully oxidized region and a value of 1.0 in the unoxidized region.

In the numerical test presented below, we consider a one-dimensional domain of length 1mm. Theleft end of the domain is exposed to air, and the simulation is run under isothermal conditions at a uni-form temperature of 288�C. Material parameters used in the simulation are given as follows: (i) the trueair density at 288�C, r

fT = 0:6273 kg

m3; (ii) viscosity of air, AL = 29:5E-6 kg=ms; (iii) gas constant,

�R = 286:987 J=kg�K; (iv) body force, bf = 0; (v) molecular weight of air is MWair = 0:02897 kg=mol;(vi) �b = 32:4412 m3

kg; (vii) oxidation state, fox = 0:187; (viii) reaction rate, R0 = 1:69E-2 kg

m3s; (ix)

true solid density, rsT = 1320 kg

m3; (x) porosity of solid, fs = 0:1; (xi) diffusivity of the solid,

D = 8:933E-13 m2 s; (xii) Young’s modulus, Es = 2:6 GPa; (xiii) k

f2 = 0; (xiv) a = 0:35 m3

kg; (xv)

�A0G = � 0:25E12; and (xvi) �K1 = � 1:0E9. It is noted that the only new parameters that are not con-strained by direct measurements are the last two parameters, that is, �A0G and �K1. The remaining para-meters are either specified in the original work [17], or are standard reported values (limited to theviscosity of air AL = 29:5E-6 kg=ms and Young’s modulus of PMR-15 Es = 2:6 GPa).

The one-dimensional domain is discretized with a graded mesh of linear Lagrange elements. The sub-set of the domain, [0,0.0001]m, is discretized with 100 elements and the rest of the domain also with 100elements . The fluid and the solid constituents coexist over this domain. A fluid concentration of22.8863E-3 kg/m3 is specified as the boundary condition for the fluid and a load of 1 atm is applied onthe solid at the left end of the domain. The problem is run with time steps of 1E-5 seconds for 1000steps, followed by a time step of 1E-3 seconds for 10,000 steps, and with a time step of 0.1 seconds for atotal time of 100 hours. The drag coefficient Av for the oxidation problem is defined in terms of diffusiv-ity of the solid, as Av = r f (r + r f )�Ru

rT mD, where m is the Lagrange multiplier. For the derivation of

this expression, refer to the Appendix.

Figure 7. Schematic representation of the thermo-oxidation process.

Hall et al. 217

Remark: In our model, the fluid properties and its initial/boundary conditions are defined in mass concentration units.Since fluid properties in Tandon et al. [17] are provided in molar concentration units, they have been converted to appropri-ate units for the present system of equations using the standard conversion relations.

The active oxidation zone that lies between the fully oxidized zone and the unoxidized core has a con-tinuous variation of f from fox to 1, respectively. Figure 8 shows the positions of the actively oxidizingdomain (Zone II, Figure 7) versus time, for four different values of f associated with a reaction rate of1.69E-3 kg/m3-s and a solid diffusivity of 8.93E-13m2/s. The upper curve f=1 depicts the position ofthe Zone III–II boundary (oxidation front) between the oxidized and unoxidized regions, and the lowercurve f=fox depicts the position of the Zone II–I boundary of the actively oxidizing zone and the fullyoxidized zone. The remaining values of f correspond to positions within the actively oxidizing domain.The oxidation layer growth results shown in Figures 9–11 are plotted for f = 0:3.

A parametric study was done for the oxidation layer growth with time and results are shown inFigures 9–11. Figure 9 shows the variation in oxidation layer growth for different reaction rate para-meters for a duration of 100 hours. The solid line shows the results from the mixture theory, where itcan be seen that the reaction rate of 2.41E-4 kg/m3-s produces an oxidation layer growth of 66.9 mm as

Figure 8. Oxidation layer growth with time for various values of the oxidation state variable.

Figure 9. Oxidation layer growth with time for various values of reaction rate.


compared to 74.7 mm for the reaction rate of 1.69E-3 kg/m3-s at the end of 100 hours. The mixture the-ory results follow a similar trend in comparison with the Tandon et al. [17] numerical results.

Figure 10 shows the growth of oxidation layer for 0.1 and 0.187 oxidation state values. Sincef 2 fox, 1½ �, the local value of f indicates the amount of polymer that is available for oxidation. An oxi-dation state value of 0.1 indicates the spatial location where almost 90% of the polymer is available foroxidation, as compared to a value of 0.187 that indicates that only 81.3% of the polymer can be oxi-dized. For a constant oxidation rate, a lower value of fox indicates that there are more sites availablefor oxidation and therefore the rate of growth of the oxidation layer will be slower, as can be seen inFigure 10. Figure 11 shows the influence of the diffusivity of the solid on oxidation layer growth inPMR-15 resin. It can be observed that a diffusivity value of 1.667E-12m2/s advances the oxidation layerat a higher rate in comparison to the lower diffusivity values of 1.3E-12 and 8.933E-13m2/s. The oxida-tion layer depths of 74.7, 90.1, and 100.1 mm are observed for solid diffusivity values of 8.933E-13,1.3E-12, and 1.667E-12m2/s at the end of 100 hours, respectively.

Tandon et al. [17] studied the oxidation layer growth via the diffusion-reaction equation assuming anideal fluid permeating through a rigid solid. Accordingly, in their model the deformation of the solidand viscous effects in the fluid are neglected. In the present work where we employ the mixture theory, aNewtonian fluid and an elastic solid are considered. Since the unknown fields in the mixture model are

Figure 10. Oxidation layer growth with time for various values of oxidation state.

Figure 11. Oxidation layer growth with time for various values of drag coefficient.

Hall et al. 219

fluid density, fluid velocity, solid displacement, and solid density, the kinematic and the force measurescan be readily obtained from the simulations. Figure 12 shows the variation of the fluid and the solidkinematic and force quantities for solid diffusivity values of 8.93E-13, 1.30E-12, and 1.67E-12m2/s. Theplots shown are obtained for a saturated oxidation state value of fox =0.187 and a reaction rate of1.69E-3 kg/m3-s. Figures 12(a) and (b) show the variation of solid density and fluid density along thedomain at the end of 100 hours. Full oxidation of all presumed available sites results in a fixed solid den-sity, as indicated in Figure 12(a).

Since there are only two constituents in the present model, loss of mass from one is the gain in massfor the other. Consequently, the density of the solid increases as shown in Figure 12(a) wherein theapparent solid density has a higher value as compared to the neat resin region. This is rather contradic-tory to the experimental observations, as the density of the PMR-15 resin is expected to decrease withincreased levels of oxidation. (It does, however, correspond to initial weight gains in certain oxidizingmaterial systems before substantial mass loss to the environment occurs. The transfer of mass out of thematerial system is not explicitly addressed here.) If the two-constituent mixture model is extended to thethree-constituent model where the third constituent is allowed to evolve and also leave the domain, itcan account for the experimentally observed weight loss in solid due to the oxidation process. Figure12(c) shows that the variation in fluid stress is dominated by the hydrostatic pressure. Figure 12(d)

Figure 12. Fluid and solid kinematic and force quantities along the domain at the end of 100 hours. (a) Solid density along thedomain. (b) Fluid density along the domain. (c) Fluid stress along the domain. (d) Interactive force along the domain.


shows the distribution of interactive force between the diffusing fluid and deforming solid. It can beseen that the interactive force becomes zero in the neat resin region where the fluid has not reached yet.

5.3 Slurry infiltration problem

SI is an important step in the processing of ceramic matrix composites (CMCs). In the SI process, a vis-cous fluid that is laden with particles of various sizes, composition, and volume fraction is injected intoa fiber preform, wherein fluid primarily serves as a medium that carries the suspended particles to thepreform. This cycle is repeated several times until the density of the preform increases and its porosityreduces to some desired design value. Once the SI process is complete, a second process called melt infil-tration is carried out with a viscous fluid that can chemically react with the preform as well as the depos-ited particles to make a composite with the desired strength and density distribution [26].

In this section, we consider SI and employ properties of a porous PMC as a surrogate model forCMC material. We assume that water-based slurry has permeated the porous elastic solid and we modelthe process of deposition of suspension onto the fiber preform. Young’s modulus of the porous PMC isobtained via the rule of mixtures as given below:

EL = Ef Vf + EmVm ð66Þ

where Ef , Em are the fiber and epoxy Young’s moduli, respectively, and are assigned values of 380 and3.45GPa. Vf , Vm are the volume fractions of the fiber and the matrix in the porous composite. For a50% porous PMC, we evaluate the properties based on 40% fiber and 10% matrix composition. Thecarbon fiber density and the matrix density are 1950 and 1200kg/m3, respectively. The water-basedslurry is assumed to contain SiO2 particles of dimension 2–15 mm with 50% volume fraction. The visc-osity of the slurry can be computed from Einstein’s equation [27] as follows:

msl = mw(1 + 2:5fSiO2) ð67Þ

where msl is the viscosity of slurry, mw is the viscosity of water, and fSiO2is the volume fraction of SiO2

particles in the slurry. Assuming 50% volume fraction of SiO2 particles, the slurry viscosity turns out tobe 1.793E-3 kg/m-s. Given that the density of the SiO2 particles is 2650 kg/m

3 and the density of water is1000kg/m3, slurry density can be computed as

rsl = 0:5 3 rw + 0:5 3 rSiO2

= 1825kg m3

ð68Þ

where rsl, rw, and rSiO2 are the density of the slurry, water, and the SiO2 particles, respectively. In thepresent model, it is assumed that the particle-laden fluid is uniformly present in the domain and thedependence of the rate of deposition on the flow velocity is ignored. Accordingly, the mass depositionrate of particles from the slurry onto the porous composite, as given in [24], is modified for the presentcase as follows:

_m f = �kr f w ð69Þ

where k is the filtration constant and w is the apparent mass fraction of the particles in the slurry. Thefiltration rate of the solid medium is assumed to be 83.8341E-3 s21. The initial apparent mass fractionof particles in the slurry can be computed as

w0 = fsVpsl

rp

rslð70Þ

where fs is the solid porosity (fs = 1� rs

rsT , rs

T is the true density of the solid) and rp, Vpsl are the den-

sity and volume fraction of particles in the slurry, respectively. The drag coefficient Av for the SI (per-meation) problem is defined in terms of permeability of the solid K and the viscosity of the fluid AL asAv = AL

K. (For the derivation of this expression, see the Appendix). The permeability of the solid is

Hall et al. 221

Figure 13. Mixture constituents kinematic and stress measures along the domain at 30, 60, and 90 seconds. (a) Fluid density alongthe domain. b) Solid density along the domain. (c) Hydrostatic stress in the fluid along the domain. d) Axial stress in the solid alongthe domain.

Figure 14. Reduction in solid porosity with time.


taken to be 4.935E-17m2. The chemical reaction and solid strain coupling parameters are assigned to be�A0G = � 0:25E3, �K1 = � 1:0E0.

In this problem, a one-dimensional domain of length 0.4m is considered that contains both solid andfluid constituents that are uniformly present everywhere. We assume uniform material properties andtemperature distribution. In addition, we assume that deposition of the suspended particles is occurringthroughout the domain. The problem is run for 90 seconds with a time step of 5E-4 seconds. The soliddisplacement and fluid velocity are constrained at the left end of the domain. A load of 1E7N is appliedat the right end of the domain.

Figures 13(a) and (b) show the reduction in apparent fluid density and increase in apparent soliddensity along the domain at the end of 30, 60, and 90 seconds. The initial apparent fluid density of912.5 kg/m3 drops to 514.9 kg/m3 at the end of 30 seconds and further drops to 393.4 and 337.1 kg/m3

at the end of 60 and 90 seconds, respectively. This drop in fluid density is due to particle deposition onto the porous solid that results in an apparent solid density increase (see Figure 13(b)) from 900kg/m3

to 1297.7, 1419.1, and 1475.5 kg/m3 at the end of 30, 60, and 90 seconds, respectively. In order to evalu-ate the evolution in the stress-carrying capacity of the solid, an external load is applied to the solid,which is held constant in time, that is, the solid is under constant compressive stress of 10MPa through-out the process. Figure 13(c) shows the hydrostatic stress in the fluid, and Figure 13(d) shows the solidaxial stress profiles along the domain. Since the rate of deposition of particles is constant along the

Figure 15. Reduction in solid porosity with time for 30%, 40%, and 50% SiO2 particles in the slurry.

Figure 16. Reduction in solid porosity with time for 40%, 50%, and 60% initial solid porosity.

Hall et al. 223

domain, the fluid and solid stresses remain constant along the domain at a given (but otherwise arbi-trary) time level.

Figure 14 shows the decrease in solid porosity as a function of time. For a 50% initial solid porosityand with a 50% particle slurry, the maximum reduction in porosity is bounded by 0.25. As can be seenfrom Figure 14, the solid porosity asymptotes to 0.25 with time.

Next, we present the results for the case where the porous solid is subjected to three infiltration cyclesof 30 seconds each, for a total of 90 seconds. At the end of each cycle, the particle mass fraction w is resetto the initial particle mass fraction in the slurry w0. Figure 15 shows the variation of the solid porositywith time for 50% porous solid and 30%, 40%, and 50% SiO2 particle volume fraction in the slurry. Wesee that as the particles get deposited, the porosity of the solid decreases. For all three different particlevolume fractions in the slurry, this decrease in porosity is nonlinear, wherein the rate of reduction in por-osity seems to be slowing down with time, which is indicated by the relatively flatter portion of the curveat the end of each cycle. From the perspective of the physics of the problem this means that while thereis more relative reduction in porosity during early infiltration cycles, because of the closure of pores thathappens due to the solid mass buildup, the relative reduction in porosity in subsequent cycles also slowsdown. Figure 16 shows a similar trend in reduction in porosity with time for three different initial solidporosities that are infiltrated with 50% particle slurry.

6. Conclusions

We have presented a VMS-based finite element method [13,15,16] for the fluid–solid mixture theorymodel of Hall and Rajagopal [12] that is based on the constituent equations of motion and mass bal-ance. The model addresses the energy and entropy production equations through an equation for theLagrange multiplier that results from consideration of the full set of balance equations as a constraintduring the process of maximization of entropy production. The present system of equations is appliedto isothermal processes in the one-dimensional context. Employing VMS ideas, a multiscale decomposi-tion of the fluid density field into coarse and fine scales and a priori unique decomposition of the admis-sible spaces of functions leads to two coupled nonlinear problems termed the coarse-scale and the fine-scale sub-problems. The fine-scale solution is extracted from the nonlinear fine-scale sub-problem,which is then variationally projected onto the coarse-scale space, leading to a formulation that isexpressed entirely in terms of the coarse scales. Although the final formulation does not depend expli-citly on the fine-scale density field for the fluid, the effects of fine scales are consistently represented viathe additional residual-based terms, and they add to the stability of the numerical method. The resultingstabilized method for the mixture model is applied to hyperbolic propagation while recovering Fickiandiffusion, anisotropic oxidation in composite materials recovering the data of Tandon et al. [17], andmass deposition. Results of the oxidation modeling of Tandon et al. [17] are recovered by employingthe reaction kinetics model and properties assumed therein; the only additional assumed properties aretwo constants describing coupled chemomechanical and purely chemical dissipation. In all of these casesthe mixture provides rich detail concerning the kinematic and kinetic behaviors of the constituents, incontrast to standard effective media approaches. The proposed solution scheme based on a singleHelmholtz energy reveals the importance of an effective material property related to the temperaturerate; further investigation in the three-dimensional context is needed to determine applicability to gen-eral anisotropic and anisothermal problems.

Funding

Partial support for this work was provided by AFRL under Contract No. FA8650-13-C-5214. This support is gratefully

acknowledged.

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Appendix – Relation between solid diffusivity and drag coefficient

In Section 5, we have presented a reduced-order mixture problem, oxidation of PMR-15 resin, and SIproblem. A literature review reveals that the reduced-order mixture problem and the oxidation problemis in general modeled via a diffusion-reaction equation, while the SI problem is typically modeled via aDarcy equation. In the context of the mixture theory model, the fluid–solid interaction is accounted forvia the interactive force field, which requires the specification of drag coefficient Av. The relationbetween the drag coefficient Av and the diffusivity of the solid D can be obtained by comparing the mix-ture theory equations and Fick’s diffusion-reaction equation. Similarly, the relation between the dragcoefficient Av and the permeability of the solid K can be obtained by comparing the mixture theoryequations and the Darcy equations for the SI problem.

Hall et al. 225

A1. Fick’s diffusion-reaction equation

Fick’s diffusion-reaction equation can be written as

∂r f

∂t= D

∂2r f

∂x2+ m f ð71Þ

where D is the solid diffusivity.

A2. Darcy equation

The fluid balance of the mass equation and Darcy’s law are given as follows:

uf1 = � K

AL

∂p

∂xð72Þ

∂r f

∂t+∂(r f v

f1 )

∂x= m f ð73Þ

where uf1 is the filtration velocity, K is the permeability of the solid, and AL is the viscosity of the fluid.

Assuming that the pressure of the fluid follows the ideal gas law, p = r f �Ru, Equations (72) and (73) canbe combined as follows:

∂r f

∂t� K �Ru

ALfs

∂

∂xr f ∂r f

∂x

� = m f ð74Þ

where fs is the solid porosity. Equation (74) can be written in an expanded form as

∂r f

∂t� K �Rur f

ALfs

∂2r f

∂x2� K �Ru

ALfs

∂r f

∂x

� 2

= m f ð75Þ

A3. Mixture theory

The fluid balance of mass and linear momentum are given as

∂r f

∂t+∂(r f v

f1 )

∂x= m f ð76Þ

∂Tf11

∂x+ I

f1 = 0 ð77Þ

where the fluid body force and inertial effects are neglected.

A3.1 Reduced-order mixture problem

Consider the constitutive relations for the fluid stress and interactive force as given in Equations (57)and (58). Substituting these constitutive relations into Equation (77), the fluid velocity can be written as

vf1 = �

�Ru

rf Av

∂rf

∂xð78Þ

Equations (76) and (78) can be combined to give

∂r f

∂t�

�Ru

Av

∂2rf

∂x2= m f ð79Þ

Comparing Equations (71) and (79), the drag coefficient can be written in terms of solid diffusivity as


D =�Ru

Avð80Þ

A3.2 Oxidation and slurry infiltration problem

Consider a simplified form of the constitutive relations for the fluid Equations (31) and (32), as givenbelow:

Tf11’� r f r

∂c

∂r f’� r f r

rT

�Ru ð81Þ

If1 ’� mAvv

f1 ð82Þ

Substituting the above Equations (81) and (82) into Equation (77), the fluid velocity can be written as

vf1 ’�

�Ru

mAv

∂

∂x

r f r

rT

� ð83Þ

Fluid velocity given in the above expression is substituted in the fluid balance of mass, Equation (76),and is written as follows:

∂r f

∂t�

�Ru

mAv

∂

∂xr f ∂

∂x

r f r

rT

� � = m f ð84Þ

∂r f

∂t�

�Rur f

mAvrT

(r + r f )∂2r f

∂x2+ � � � = m f ð85Þ

Comparing Equation (85) and Equation (71), we can obtain the following relation for solid diffusivityand drag coefficient for the oxidation problem as

D =�Rur f

mAvrT

(r + r f ) ð86Þ

Comparing Equation (85) and Equation (75), we can obtain the relation between the drag coefficientand the permeability of the solid for the SI problem as

Av =ALfs

KmrT

(r + r f ) ð87Þ

In the Section 5.3, we have presented numerical results for a simplified form of the above relation,

Av’AL

K.

Hall et al. 227

mathematics and mechanics of solids diffusion of...

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