Modelling matrix injection through elastic porous preforms

ELSEVIER PII: S1359-835X(97)00029-8

Composites Part A 29A ( 1998) 5- I8 0 1997 Elsevier Science Limited

Printed in Great Britain. All rights reserved 1359-835X/98/$1 9.00

Modelling matrix injection through elastic porous preforms

D. Ambrosia and L. PreziosibT* aCRS4: Centro Ricerche e Sviluppo Studi Superiori Sardegna Via Nazario Sauro 10, Cagliari, 09 123, Italy bDipartimento di Matematica, Politecnico, Corso Duca degli Abruzzi 24, Torino, 10129, Italy (Received 10 November 1996; accepted 17 January 1997)

This study deduces a new model aimed at simulating injection moulding processes under isothermal conditions and in situations in which infiltration generates deformations in the reinforcing network, both in the solid preform part which is still dry and in the part that has already been infiltrated. As a by-product the model allows the stress and deformation states in the solid preform to be monitored. The coupled flow/deformation problem in the fully infiltrated region and in the uninfiltrated region is then formulated with the proper evolution equations for the boundaries delimiting the two domains and with the relative boundary and interface conditions. The stationary configurations for the one-dimensional problems characterized by given infiltration velocity or pressure are analytically determined for general elastic stress-strain and permeability-volume ratio relations. The unsteady problem is solved numerically, putting in evidence the influence of the deformations on the bulk flow and on the propagation of the advancing front. 0 1997 Elsevier Science Limited.

I Keywords: analytical modelling; injection moulding; resin transfer moulding CRTM))

INTRODUCTION

Many composite manufacturing processes, such as resin transfer moulding, structural resin injection moulding and squeeze casting, can be schematized as infiltration problems through an initially dry porous material.

In modelling these processes, in the literature, the solid preform is usually assumed to be rigid, although several papers ‘L” show photographs or qualitatively describe deformation of the solid preform (see also Ref. i4 for a recent review of the subject).

In fact, in several practical situations the preform is far from being rigid and the pressure gradient driving the flow can generate significant deformation, especially ahead the advancing infiltration front and where the liquid matrix penetrates the preform. Of course, the compression of the preform decreases its permeability hampering the flow. For this reason deformations need to be taken into account in simulating composite materials manufacturing. Further- more, from an industrial point of view, it is important to monitor deformation and stress states in the solid preform to identify in advance possible inhomogeneities and damages in the reinforcing network.

*Corresponding author.

Of course, higher infiltration pressures cause stronger deformation and, vice versa, deformations could, in principle, be controlled reducing the applied pressure. There are, however, several reasons for which a sufficiently high pressure need be applied:

(1)

(2)

(3)

(4)

The mould has to be completely filled before the resin gel cures, or the metal melt solidifies. The entry of metal melts is usually opposed by the necessity of generating meniscus curvature at the infiltration front. It is, therefore, necessary to overcome this adverse capillary pressure before infiltration can even start. A higher applied pressure reduces the infiltration time, which, in turn, reduces the occurrence of undesired interfacial reactions and degradation of the fibrous reinforcement. From the industrial point of view it is always desirable to obtain the final product in the shortest time possible, but still in good shape.

In order to determine a satisfactory compromise among the different aspects, one has then to model the whole infiltration process, in order to identify the optimum set of operating parameters and predict a window of applicability in the parameter space as qualitatively attempted by several authors ‘,‘6,‘7.

5

Modelling matrix injection through elastic porous preforms: D. Ambrosi and L. Preziosi

This paper will deduce a model aimed at simulating injection moulding processes under isothermal condition, dropping the usual assumption that the solid preform does not deform during infiltration. A one-dimensional simulation is performed and its results are compared with some experimental results lo toward the study of the validity of the model to describe those situations in which deformation of the reinforcing network cannot be neglected, as observed by several authors ‘-14.

The problem then presents the formation of three time- dependent domains, the first one occupied by the liquid only, the second one by the solid preform wet by the liquid matrix, and the third one consisting of the uninfiltrated solid preform. It is assumed that a sharp front divides the three domains. If this might be plain for the interface dividing the pure liquid and the wet preform, i.e. one of the borders of the preform, it implies a simplification of capillary phenomena at the interface between the saturated and the dry porous medium. This assumption, often called slug-flow approximation, is reasonable when the applied pressure is much larger than the capillary pressure.

Another non-trivial problem which will be dealt with is the deduction of the evolution equations for the interfaces delimiting the different domains and the formulation of the relative boundary and interface conditions that need be joined to the deformable porous media model.

The stationary configurations for given infiltration velocity or given pressure are analytically determined for

general nonlinear elastic stress-strain and permeability- volume ratio relations. A relation between maximum deformation of the preform and the work parameters (namely, preform dimension, stiffness, and permeability, melt viscosity, and infiltration velocity) is also determined.

Finally, the unsteady coupled flow/deformation problem, which, under elasticity assumptions has a hyperbolic structure, is solved by finite differences methods using upwind methods based on characteristic decomposition.

Section 2 of this paper presents the isothermal model based on the theory of deformable porous media suitable to simulate squeeze casting, or injection moulding processes, in which deformation of the solid preform cannot be neglected and needs to be monitored. In Section 3 the evolution equations of the interfaces delimiting the time- varying domains and the relative interface and boundary conditions are determined. In Section 4 the steady problem is analytically solved. Finally, Section 5 deals with the numerical method, discusses the results of some tests of infiltration processes in an initially dry and elastic porous preform and compares them with some experimental results” obtaining a satisfactory agreement.

MODELLING THE INFILTRATION PROCESS

Squeeze casting and injection moulding manufacturing processes can be schematized as infiltration problems of

Figure 1 Geometrical schematization of the infiltration problem

6

Modelling matrix injection through elastic porous preforms: D. Ambrosi and 1. Preziosi

incompressible liquids in deformable porous media made up of an incompressible solid constituent.

In the Appendix, the steps used in a previous paper *’ to deduce a three-dimensional model which allows deformation of the solid network and its one-dimensional specialization are briefly recalled. Here, we will start considering the following set-up.

The dry preform occupies for t < 0 the region x E [0, Ll and is compressed at a given volume ratio $a,(~,r=O)= &+), or possibly it is in its undeformed configuration Q?I~(x, t = 0) = $r.

The right border of the solid preform is constrained by a net which prevents the preform from being pushed beyond .x=L, but allows, at the same time, both air and liquid matrix to pass through with no resistance (see Figure 1).

The liquid matrix flows in the positive direction either pushed at a given velocity, or forced by a pressure gradient. At t = 0, it touches the free border x = 0 of the solid preform.

The incoming liquid compresses the sponge and starts infiltrating. Therefore, while the right border of the sponge is fixed at x = L, the other one moves to .r = .x~([), and part of the preform wets, say up to x=.x,(t).

In doing this we are implicitly simplifying capillary phenomena assuming the existence of a sharp flat front, which divides the fully saturated porous medium D”’ = [xf(t),x,(t)] from the remaining uninfiltrated portion Dd = [-vi(t), L]. This assumption, often called slug-flow approximation, is valid when the applied pressure is much larger than the capillary pressure.

As infiltration proceeds both the dry and the wet preform compress or expand back, according to the flow conditions. However, as will be shown in the Section 4, even long after the liquid matrix has reached the other end of the sponge x = L, the final steady configuration (which can be reached if the liquid is steadily pushed into the channel) will be non-uniform. The undeformed configuration can be reached again only if liquid inflow is stopped and further relaxation of the sponge with spontaneous imbibition is allowed.

Referring to the Appendix, one can write for the wet part of the sponge D”’ = [xf(f),xj(r)l the following model

(1)

f?dJr ( d Vi F+“:‘g =__ 1 &v: - %dt)l + 2, (2) J

where the superscript w stands for wet, ps is the density of the solid material used to make the preform, 4,$ is the volume fraction of the solid preform, i.e. the volume occupied by the constituent over the total volume, v,? is its velocity, p is the matrix viscosity, Uin is the inflow velocity, and 7: is the excess stress of the solid-liquid mixture.

Eqns (1) and (2) could also be applied to describe the motion of the dry preform in Dd = [x;(t), L] and the flow of

air in it, of course with the appropriate changes (p replaced by the viscosity of air plLir, and the superscript w by d, standing for dry).

However, viscosity of air is so small that it is easily expelled from the dry preform offering negligible resistance and contribution to the stress of the mixture. This means neglecting the first term on the r.h.s. of eqn (2) and approximating r2Z with rf. One can then write

(4)

As already stated in the Appendix, owing to the scarcity of experimental data on the viscoelastic properties of both the wet and dry preform, in this paper it will be assumed that they behave elastically

T:/ = - C,J4f), in D”. (6)

with C,,, and C, strictly increasing functions of the volume ratio. Under this assumption the system of eqns (l)-(4) is hyperbolic.

We finally observe that most papers in the literature measure C,5(4,5) and assume. more or less implicitly, that

&(6., I= L(~,). (7)

which means assuming that the response of the porous material is independent on its being dry or wet.

This step can be considered an approximation, since from a continuum mechanics viewpoint. C,,, contains other contributions besides C,, such as the stress in the liquid and a fluctuation term “. Besides. from an experimental viewpoint, several authors have shown that the static response of a porous medium depends on whether it is dry or wet. For instance, Beavers and coworkers ” 33 explicitly observe ‘a reduction in the compressive force when a specimen is completely saturated and submerged in water for all polyurethane foams (. . .) used in the experiments. Although the mechanism underlying this behaviour cannot be identified with certainty. it is believed that the liquid (i.e. water) tends to lubricate the deforming solid and therefore diminishes the resistance to deformation.’ This is also a matter of common experience, when squeezing some sponges before and after dipping them into a liquid.

INTERFACE AND BOUNDARY CONDITIONS

In order to close the system, we have still to add proper interface and boundary conditions to eqns (l)-(4) and give the evolution equations for the interface x,(r) and the free boundary q(r).

The interface x =xi(t) dividing the domains is a material

7


interface fixed on the liquid phase, and, therefore, has to move with the velocity of the liquid at the interface. Still referring to the Appendix, and, in particular, to eqn (A.16), one then has

&i Uin(t) - 4Y(xi(r)9 t)vl(xi(t), t,

dt= 1 - 4Y(xiCt>7 t) ’ (8)

On the contrary, the sponge border x=xf(t) is a material interface fixed on the solid phase, and, therefore, has to move with the velocity of the solid at the interface. Hence

dxf = V”(Xf@) t) dt ’ ”

This also reduces the stress interface condition (eqn (A.14)) to a stress continuity condition across q(t). Since the liquid pressure is continuous across the interface as well, then eqn (A.14) simplifies to

C(X#, t) = - Ll(@G,(r), t)) = 0, (10)

that is the free border of the preform is locally in the undeformed state

d,“(#)Y r) = C, (11)

which is implicitly defined by eqn (10). Of course, since the other border x=L of the solid

preform is constrained by a net

&L, t) = 0. W) hi At the interface xi(t), where v, = dr= VT, eqn (A.14)

specializes to

(13)

where v; is given by eqn (A.16), and all variables are computed in (xi(t), t).

The stress condition, instead, writes

The interface conditions eqns (13) and (14) are not in closed form as eqns (11) and (12), because they couple the solution between the domains. They correctly obey to the character- istics theory, as the systems of eqns (l)-(2) and (3)-(4) require one boundary condition each.

Substituting eqn (A. 16) evaluated in xi(t) and carrying out some calculations, one can put in evidence the following relations

UK> - &&#4) = Ps 9f(+,” - 4$)Lv% - Uin(t)12

(1 -&y# . (16)

It has to be mentioned that the dependence of C on 4, which must be deduced from experimental measurements, must in general be such that the relationships (15) and (16) are single-valued. If, for instance, it is assumed that the wet

and dry porous material behave elastically and in the same way (i.e. Eqn (7) holds), one has that

v:($(t), t, = vf(xi(t), t), (17)

4P(xi(t), t, = d$(xi(t), t), (18)

which means that both the volume ratio and the velocity of the solid constituent are continuous across xi(t).

As already mentioned, however, at the end of the previous section this is not genuinely true, since different constitutive models should be used in the two domains. Using the same constitutive model and the same stress- strain relation in the two domains, and therefore (eqns (17) and (1 S), has to be considered an approximation.

The initial boundary value problem is then given by eqns (l)-(4) joined with the evolution equation for the interfaces in eqns (8) and (9), the boundary and interface conditions of eqns (1 l)-( 14) (we remind that the quantities appearing in eqns (13) and (14) are evaluated at (Xi(t), t)), and the initial conditions

Xi(O) = 0,

if Xi(t) < L, that is for t < v if Uin is constant. When the solid preform is fully infiltrated, i.e. Dd = 0,

then the integration of the initial boundary value problem continues with eqns (1) and (2) with the evolution eqn (9), and the boundary conditions (11) and

vY(L, t) = 0. (20)

The quantity Uin(t) which appears in the initial boundary value problem depends on how the liquid matrix is injected in the solid preform. If the liquid is pushed by a piston at a given velocity, then Uin(t) is actually a given function of time, e.g. Uin = const.

A more interesting situation arises when the flow is driven by a prescribed pressure difference

Ape(r) = M+(t), t) - Pe(L t), (21)

applied between the extrema of the solid preform (say, constant), which therefore will determine the inflow velocity through Darcy’s law. In fact, exploiting the fact that in one dimensional problems the composite velocity is constant (see Appendix), one can explicitly write the velocity of the constituents as

V,” = Uin(t) + K(K) ape

-dx P (22)

or

K($d> ape Vf = Uin(t) + sp

Pair dx ’ (23)

in Dw; (24)

[v~(x, t) - ui,(t)] in Dd.


Integrating eqn (24) over [am, L] gives and eqn (11) can be rewritten as

&(f) L AP&) = - p

s

$(.X1 t)dX

s

&X, t)dX

If(r) K(4Y(t, X)) - Pair x,(t) K(&t,x))

[I

-t,(t) dx + uiR(f) P .1,(t) K(@(t,x))

J L dx + Puir r,(r) K(g!(t,x)) I ' (25)

which can be explicated with respect to uin as

Since the viscosity of the gas is much smaller than the viscosity of the infiltrating liquid, then the pressure drop in Dd is negligible, which reduces eqn (26) to

Ape(t) + J

4(r) vY(x, t)dx

L&(l)= p V(f) K(#(t,x)) C-r,(l) & .

(27)

J .q(O K(#yt,x))

However, in our problem initially xi(O) =xf(0) = 0 and therefore at the very beginning x;(t) and xf(t) are very close. At this early stages of infiltration one should then consider the full relation of eqn (26), instead of eqn (27). In fact, for

the two integrals at the denominator may be of the same order of magnitude.

Using standard work parameters this occurs at most for t < 0.001-0.01 s yielding an initial inflow velocity of the order of one meter per second.

STEADY STATE AND ANALYTIC SOLUTION

Assume that the input velocity Uin(t) of the liquid, or, equivalently, the pressure gradient driving the flow, tends to a constant. In this situation one may look for stationary solution in which the solid preform is all wet and at rest at a volume ratio 4:(x, t) = 4:(x). In order to describe this state, we will drop, for sake of clarity, the superscript w, observing that in this section all variables are time independent.

From eqn (2) we can then write

where

E’(k)= g> 0, J

(29)

Mf) = 4,. (30)

Eqn (30) does not provide a proper initial condition to solve the ordinary differential eqn (28), since it gives the volume ratio (4, = 4,) in a location x = xr to be determined. On the contrary, the volume ratio at the fixed end of the solid preform and the length of the compressed preform are unknown. As we shall see, the mass of the solid preform, which is constant throughout the evolution of the system, will furnish the necessary information to solve the problem.

Integrating eqn (28) gives

where, as already mentioned, 4,JL) is still to be determined. Given a volume ratio 4, E [4,., 4,(L)], eqn (31) gives the

locations where it is assumed

In particular,

1 @,(L) -9 =L - z 1. $ K($)C'($W. (33)

The value d,(L) which appears in eqns (31)-(33) can be obtained recalling that the mass of the sponge is preserved during the evolution. In particular, it is equal to the mass at time t = 0

J L

hr. = x k(x)~. (34) I

If C(d,) is strictly increasing, then from eqn (3 1) also ~,Jx) is strictly increasing. Transforming the integral over .r in an integral over 9, gives

J L d,(L) Xl Mm+ i J. 44,Mf, + 4r.q = 4?s(LL (35)

or, using eqn (34) and then eqn (32).

Changing order of integration in the double integral one can write

(37)

9

Modeling matrix injection through elastic porous preforms: D. Ambrosi and L. Preziosi

Using eqn (33), eqn (37) can be simplified into

s

9*(L) P”in4rL = ~ W4,)C’(4,)4,d4,, (38)

,

which is an integral relation implicitly defining 4,(L). Once K(4,) and X(4,) are given, and 4,(L) is computed

using eqn (38), the steady solution is given by eqn (32) and, in particular, the location of the free border by eqn (33).

It has to be observed that, since the integrands are all positive, 4,(L) and xf are increasing functions of uin, This means that, unless uin = 0, or, equivalently, APe = 0, xf > 0 and the preform is neither relaxed nor uniformly compressed. As expected, however, if the viscosity or the inflow velocity are very small, or if the solid preform is very stiff, then the steady configuration of the preform is nearly undeformed.

As an example we will now consider the permeability- volume ratio relation and the stress-volume ratio relation measured by Sommer and Mortensen lo for a polyurethane sponge (TF-5070-lo), namely

K(4,) = 1.4729.10- 10-6.96549s m2 (39)

with

a0 = - 23.29164, al = 713.49318,

a2 = - 9255.285, a3 = 65974.87,

a4 = - 279182.3, a5 = 701653.83,

a6 = - 970226.7, a, = 569801.8,

(see Figure 2), which, in particular, yield the following values in the undeformed configuration

4, = 0.135175, C’(4,) = 2.522532 MPa

and K($,.)= 1.6851.10-” m2.

This choice is simply owing to the intent of using the time dependent experimental data reported by Sommer and Mortensen lo for a comparison (in Section 5). Of course, more proper relations referring to the materials used to fabricate composites can trivially replace our choice.

Figure 3 gives the stationary configuration 4,(x/L) at different values of the dimensionless parameter

s = C’(4,)U4,) PUinL ’

(40)

Therefore, higher values of S correspond, for instance, to stiffer, or more permeable solid preforms, less viscous fluids, smaller inflow velocities.

Clearly, the smaller S is, the more inhomogeneous the steady configuration is.

In Figure 4 the ratio of the position of the preform border over its undeformed length, i.e. x/L, and the value of the volume ratio at the fixed end of the sponge x = L (which, in the stationary configuration, is the maximum preform deformation) are plotted versus S. Again the higher ._Y is, the less deformed the solid preform is. In particular, Figure 4 and the original eqns (33) and (38) give an indication of the

400

3 B I g co

200

0 I I> __

0.10 0.20 0.30 volume ratio

Figure 2 Stress-volume ratio relation as measured by Sommer and Mortensen ” for a polyurethane sponge

0.50

0.40

.P 3 E 0.30 3 B

0.20

//I S=lO

, s=20

0.10 1 I I I ’ I - 0.0 0.2 0.4 0.6 0.8 1.0

X/L

1 I

Figure 3 Stationary state. Volume ratio versus x/L at different values of s = E’(I,Yw,)

T

10


0.8 /-

0.6

_ - --I-(-- and, in particular,

\

\\. volume ratio ‘\ --~_-.-__~_ \

material boundary ---____ %-__

S

Figure 4 Stationary state. Dimensionless position of the mat .eri al boundary x /L and volume ratio 6,(L) at the fixed boundary as a function of s = x (4,4W)

IlU,,,L

_i

work parameters that can be possibly used to control preform deformation.

The case in which the pressure gradient APe is imposed and eventually constant is somewhat different, but easier. In fact, from the stress equilibrium equation, one directly has

C&(X = L) = C - ’ (Ape), (41) which can be used as ‘initial’ condition to solve the ordinary differential equation. However, the constant Uin in eqn (28) is not given explicitly, but depends on the dimension and the deformation of the solid preform. However, as before, the fact that the total mass is given determines

which can be substituted in eqn (32) and (33) to have

s

Em’(AP,)

I

z+l(Ap )

K(II/)C’($)&

(43)

SIMULATION METHOD AND RESULTS

1. (44)

As already mentioned in Section Section 2. the system of eqn (l)-(4) to be integrated is hyperbolic. It turns then useful to write it in conservation form, introducing the mass fluxes qr = c#I~v,~, and qt= 4:‘v:‘, and the dimensionless variables obtained scaling lengths with the initial length of the preform L. velocities with a characteristic velocity I/, and time with LJU.

If Uin(f) is specified, then ZJ is its order of magnitude hw. If, instead, Apt(t) is specified and P, is its order of magnitude, then U = d,K(&!)/,uL.

The dimensionless system of equations then writes

(45)

pg+ ” p(c?:)2 at ax [

-+sr,,co:,] = +~~[“““t’- 51, 4:

(46)

(47)

1 =o. (48)

where now x, t, 4% and qyare dimensionless variables,

with

and

(ui,(t>

I n

4n

Ape(t) x1(r) iii,(t) = -+

AP, I v:(x, t)dx v%, t)h -VW B(qJ,W(t, x))

+&r l p I x,(t) zl(&t, x))

x,0) dx 1

-go &#J~(t,x)) I Pair dx

p 40) &f&,x))

if Uin(t) is given

if Pe(t) is given

(50)

11


As explained at the end of Section 3 after the very first steps interval, while the N + 1 numerical fluxes fi, i = 0, . ..N. are of integration the integrals over [pi, L] in eqn (50) can be defined at the border of the intervals. This allows to impose dropped, since the pressure difference across the dry naturally the boundary conditions on the value of the fluxes, preform becomes negligible. as discussed in Section 3.

The coupled flow/deformation problem (eqns (45)-(48)) depends on two dimensionless parameters

Referring to the discussion at the end of Section 2, in the simulation we assume that the mechanical properties of the wet and dry solid preform are the same, which imply continuity of volume ratio and velocity at the interface xi(t) dividing the domains (see eqns (17) and (18)).

Psww:‘) PL

if am” is given;

P= Wd) 2

P,A& 2 [ 1

(51)

CLL if Pe(t) is given;

and

if uin(f) is given;

if Pe(t) is given.

(52)

The remaining equations and initial, interface, and boundary conditions remain formally unchanged.

Eqns (45) and (46) can be formally written as

au af z+ ~=sw, (53)

where

u=(4,Pq),f = (q,p;+sE),s= (o, “n;;; “). (54)

Eqns (47) and (48) are formally the same, but for the van- ishing of the source term s.

For numerical purposes, as the integration domain stretches in time, it is useful to consider the integral formulation of the equation above, because at a discrete level it allows to take easily into account of the movements of the grid nodes. The integral formulation of the conservation law (53) reads 26

a

I

X2(f) udr+; x2(0 at Ss dx x1@) XI 0 X2(f) (55) = J x,,sw? wx,w, X*(G) c Gy WY xi(O)-

where u, s have already been defined, and f, defined now as

f = (+$,P@-v’) +SL’), (56)

takes into account of the movement of the extrema of the interval (xl(t),x2(t)).

This is the form of the flux vector when considering possibly moving domains with velocity v’(x, t). The contribution of the flux due to v’ says that the total quantity of u in the interval (xl(t),x2(t)) changes in time as it flows through the boundary of the material because of the movement of the extrema. As the whole solid preform lays in (xi(t), L) we divide this region into N equally spaced intervals; the discrete solution Ui is defined in the ith

This allows a simplification in handling the interface xi(t), which can just be treated as a marker, indicating what is the portion of the computational domain that includes a source term in the equations. Conversely, the movement of the material boundary xf(t) requires a real stretching of the computational domain. This poses some problems, as the movement of the solid is to be taken into account, while preserving a constant distance between the discretizations nodes. This is accomplished by imposing a proper discrete mesh velocity movement v’r, i=O, . ..N. such that the discretized counterpart of eqn (55) reads

;[(,),+1,;+1 -(A+;] +fY -f;_,=(Ax)n+1S(z4"+'),

(57)

where fy depends on the solution U” and on the velocity of the grid v’r, which is defined as follows:

v’r=vi ( I

1 -i , i=O ,.... N. (58)

In this way, the convective flux at the material boundary, i.e. f o, is automatically zero and, at the same time, the intervals are moved in such a way that they all have the same length (Ax)“. It is to remark that V’i coincides with the material velocity of the solid only at the left boundary. The position of the ith point at time n + 1 is xy + v’:At and the spatial integration step (AX)“+ ’ is

(A.#+’ = (~7 + v’;At) - (x;_ , + v’;_ 1 At)

= [x;+v+ - +]

- [x:-,+$(1- $)At] =(A+v;$.

(59)

In particular, if the integration intervals are initially equi- spaced, they remain equi-spaced at all the time steps.

The source term on the right hand side of eqn (57) can be straightforwardly discretized implicitly, because it is not differential: when first computing @+ ‘, the implicit discretization of s”+ ’ can be easily performed, because s depends linearly on q, as may be seen in eqn (54).

To impose proper boundary conditions, we observe that the mass and convective momentum flux on both sides of the sponge must be null; moreover, at the left boundary the sponge is in its relaxed state, while the stress at the right boundary is unknown. Therefore the flux vector on the boundary is imposed as follows:

f;;= (O,O),f:,=(O,C(~~-I)). (60)

12


The form of the discrete flux f: that we have used in the computed by a second order upwind scheme, exploiting a interior of the domain is still to be defined. We do not go characteristic decomposition of the jacobian of the function into details about this, just mentioning that a numerical fan (see also Ref. *’ for the basic mathematical back- approach which is well suited for non linear conservation ground). This technique avoids unphysical oscillations in laws has been adopted, namely, the numerical flux is the solution, even if it has a discontinuous behaviour, as it

0.150

.; 0.145 L

s T

I

~ 0.21 0.43 I - - - 0.64

--- 0.65 i: ,I

- - steady state ,// //

.// '/

,; ‘4 //

/ 0.135 // / , / , L

0.0 0.2 0.4 0.6 0.6 1.0

0.32

.s e

s B 0.22

! I. I

I / !

x/L

1’ __ 0.4 0.59 / ---- 0.76 / - - - 0.96 / -~ - - steady state /

i i i ,

! /

0.12 ’ 0.0 0.2 0.4 0.6 0.6

x/L

/ I

1 1.0

(4

1.0

S 0.5

0.0 (

W

1.0

S 0.5

r

t

interface ,I-

// i’

i’ ,/

/

/ /’

material boundary

, 1.0 t

interface

materialboundary

3.0

Figure 5 Infiltration in an initially relaxed preform due to a constant inflow velocity corresponding to P = 4 X 10 _ ' and 5 = 1 IO in (a) and to P = 4 X IO ' and3 = 50 in (b). On the left, the volume ratio is plotted at different times. The discontinuity in slope locates the advancing front x,(t) separating the wet and the dry region. On the right, the temporal evolution of the infiltration front x,(r) and of the material boundary xl(f) is given

13


0.190

0.160

0.170

0.150

0.140

- 0.05 0. ,

---- 0.26 - - - steady state

0.130 0.0 0.2 0.4 0.6 0.6 1.0

0.32

.s e

s 7 0.22

x/L

- 0.005 0.04

---- 0.1 --- 0.36 - - - steady state

,----

0.12 - 0.2 0.4 0.6 0.8 1

-F -, 1’ I’

/ I

/

I

/

L. , ,

X/L

(a) 1 interface

“ol t 0.8

0.2

material boundary

0.0 0.0 0.5

t

U-4

1 .o

0.8

0.6

S

interface

\ material boundal

Figure 6 Infiltration in an initially relaxed preform due to a constant pressure drop corresponding to P = 4 X lo- 5 and S, = 100 in (a) and to P = 4 X 10 4 and S, = 25 in (b). On the left, the volume ratio is plotted at different times. The initial sudden compression at the preform end generates a compression front propagating into the sponge, which is evidentiated in (b) fort = 0.005. The discontinuity in slope locates the advancing front xi(r) separating the wet and the dry region. On the right, the temporal evolution of the infiltration front xi(r) and of the material boundary +(t) is given. The preform is fully infiltrated when x,(t) = 1. The oscillations of the interfaces are due to the elasticity of the preform

14


happens in the present case when the liquid starts infiltrating.

The numerical method could be controlled by computing the total mass of the solid preform

*2(t) L

M= _r,(,)P.sm~ f)dJ + I, , (,)P.d(4 tw

However, summing up eqn (57) for ,j= 1, . . . . N it is

0.33

.g E

f <

0.23

r------------

1 ;/ - ---- 0.01 0.03 ,’ i

/ ,’ I! - - - /I 0.05 0.07 1 ,“I - -- 0.09

X/L

(b)

jl_j

0.33

0.28

.g !z

E 5 0.23

__ 0.11 0.26

---- 0.42 - - - 0.59 - - steady state

0.13 -0.8 -0.2 0.2 0.8

x/L

interface

material boundar Yl

Figure 7 Infiltration in a preform initially compressed at a volume ratio &(I) = 0.3 due to a constant pressure drop corresponding to P = 4 X 10 -’ and S, = 25. (a) Volume ratio at different times. The discontinuity in slope locates the advancing front x,(t) separating the wet and the dry region. The temporal evolution of x,(t) and of the material boundary xf(r) is given in (b). The preform is fully infiltrated when x,(t) = 1. The initial sudden compression is not as strong as in Fig. 6, and therefore generates smaller oscillations

15


immediately found that the mass is exactly conserved by the present approach.

In the simulations which follow we will use, as a starting point, stress-strain and permeability-volume ratio relations similar to the one measured by Sommer and Mortensen lo for a polyurethane sponge (see eqn (39) and Figure 2), since at the end we want to compare the evolution predicted by our model with their data of a time-dependent experiment. In order to examine the possible scenarios that might present, however, in the simulations we will test the model considering several flow conditions characterized by different values of the dimensionless parameters. This corresponds to multiplying these quantities by suitable constants, still keeping the form of the constitutive relations.

Figure 5 presents a simulation obtained for given uin corresponding to P= 4 X 10e5 and S= 110 in (a) and to P = 4 X 10e4 and S = 50 in (b). The figures on the left give the evolution of the volume ratio versus x at different times ti. Those on the right give the position of the infiltration front xi(t) and of the boundary of the preform xf(t). In both cases the preform gradually compresses, and the position of the preform boundary monotonically increases. The infiltration front xi(t), which can be easily identified in the graphs on the left locating the discontinuity in the slope of the solution 4,(x, ti), travels at a nearly constant speed. It has to be remarked that the volume ratio in the dry region is nearly space independent, but depends on time. After x;(t) has reached the other end of the sponge, which, in dimensionless terms, occurs for t = 1 - & in agreement with the theoretical predictions, the solution tends to the stationary configuration determined in Section 4. The behaviors in (a) and (b) are qualitatively similar but for the fact that, as expected, smaller .Y give rise to much stronger deformations.

In Figure 6 the flow is driven by a constant pressure jump between the extrema of the solid preform. In the caption, a pedix P is added to S to avoid confusing this value with the one defined for given inflow velocities and also used in Section 4. The two values are linked, in the stationary case, by the integral relation (42), which relates the steady inflow velocity and the pressure drop.

As already stated at the end of Section 3, at the very beginning, the applied pressure difference generates a very high inflow velocity (say, up to 10 m SC’), due to the smallness of air viscosity and of the infiltrated layer. Then, the elastic preform suddenly compresses. Owing to the hyperbolic character of the evolution equation, which is in turn related to the elastic nature of the preform, a compression front, evident in Figure 6(b) for t = 0.005, propagates into the sponge at a dimensionless speed of the order of m. The speed, however, is not constant, since it depends on ??‘(d,). Owing to its elasticity, after compressing the preform expands back, generating oscillations in the location of the interfaces having period related to the time needed for the compression-rarefaction wave to travel back and forth between the borders of the preform (see the figures on the right). Referring to Refs. l4 and *’ for a more detailed discussion, it is possible to state that these

16

effects will be strongly smoothed out if the solid preform were more properly modelled as a Voigt-Kelvin solid or as an anelastic solid. In this last case the system of equations would still be hyperbolic, but a stronger attenuation factor would be present. In the mean the preform compresses and then partially expands back. It has to be remarked that even when the oscillations die away, the infiltration velocity is not constant. The time needed to achieve full infiltration can be identified locating the discontinuity in slope in the figures on the right, i.e. when the graph reaches x = 1.

Figure 6(a) refers to P = 4 X 1O-5 and S, = 100, while Figure 6(b) refers to a larger P=4 X lop4 and a smaller S, = 25, which gives rise to larger deformations. It has to be remarked that the volume ratio in the dry preform becomes nearly space independent only after a transient.

In Figure 7 the preform is initially pre-compressed at a volume ratio ~#~~(x)=0.3. This reduces the initial further compression of the preform reducing the amplitude of the oscillations. After the oscillations die out, both xi(t) and xf(t) go like m, until xi(t) reaches one. Then, of course, s(t) tends to the steady value predicted in Section Section 4. Of course, the final state is the same as in the previous case. It has to be mentioned that to simplify this last comparison, the value of P in Figure 7 is based on the undeformed length of the preform, while distances are still made dimensionless with the initial dimension of the solid preform.

Figure 8 compares our numerical results with experimental data obtained by Sommer and Mortensen lo. In the figure the infiltrated part of the preform [X~(t),xi(t)] is linearly mapped onto the fixed interval [0, l] and the volume ratio distribution after 12 and 26 s of infiltration is reported. The numerical prediction given by our model is in good agreement with the experimental data.

0,

.1 8 n 1

Figure 8 Comparison between our numerical results and the experimental data of a similar experiment performed by Sommer and Mortensen lo. In the figure the infiltrated part of the preform [x,(f),zq(t)] is linearly mapped onto the fixed interval [O,l] and the volume ratio distribution after 12 and 26 s of infiltration is reported


CONCLUSIONS

We have deduced a new model conceived to simulate, under isothermal conditions, the infiltration problems occurring in some manufacturing processes used to fabricate composites, such as squeeze casting, resin transfer moulding, and resin injection moulding, but that can also be applied with the obvious changes to compression moulding processes.

In fact, in the literature, these processes are modelled assuming that the solid preform is rigid, in spite of the fact that in several cases the preform is far from being rigid and the injection velocity or the pressure gradient driving the flow can generate non-negligible deformation. The model presented here can describe these situations and allows stress and deformation states in the solid preform to be monitored. The model can be trivially extended to include curing aspects, and viscoelastic properties of the constituents, while its non-isothermal formulation is currently under study.

The infiltration problem, which can be considered as a deformable porous media problem, consists of a coupled/ flow deformation problem in two time-dependent interfaced domains, the fully infiltrated and the uninfiltrated region. The problem is formulated with the proper evolution equations determining the positions of the boundaries delimiting the two domains and with the relative boundary and interface conditions.

The stationary problems related to given infiltration velocity or given infiltration pressure are analytically solved for general nonlinear elastic stress-strain and permeability- volume ratio relations. This, for instance, allows to determine a relation between the work parameters (namely, preform dimension, stiffness and permeability, melt viscosity and infiltration velocity) and the overall compression of the preform and its maximum deformation, of course in the steady state, which represents a starting point for a general description of the physical situations that will ultimately present.

Finally, the unsteady problem is solved by finite differences methods using upwind methods based on characteristic decomposition. A comparison of the temporal evolution of the system as predicted by our model with some experimental results ‘e is performed and yields a satisfactory agreement. I5

ACKNOWLEDGEMENTS

The authors are grateful to the Italian National Research Council (C.N.R.), to the Italian Ministry for the University and Scientific Research (M.U.R.S.T.) and to the Sardinian Regional Authorities for funding the present research.

APPENDIX

The coupled flow/deformation problem related to the infiltration of a liquid in a deformable preform can be described in the framework of the theory of mixtures, which in the

isothermal case and in absence of chemical reactions and phase changes is

(Al)

(AZ)

where ps is the density of the solid material used to make the preform, and pe is the density of the liquid matrix to be injected, +,T and +l are, respectively, the volume fraction of the solid preform and the liquid matrix, i.e. the volume occupied by the constituent over the total volume (4,V + C#I~ = l), v,, and v( are, respectively, the velocity of the solid preform and the liquid matrix, T, and TY are the so-called partial stress tensors, and m is the so-called internal body force, which describes the microscopic interactions between the solid preform and the liquid matrix across the interface separating them.

The main difficulty in using this theory is in formulating and validating the constitutive relations for the partial stresses, and the internal body force appearing in it. In particular, neither the partial stresses nor the internal body force can be tested directly. Information can only be obtained on the basis of thermodynamic arguments classically used in continuum mechanics (see, for instance Ref. lx.)

Under some assumptions, however. it is possible to deduce Darcy’s law for the infiltration of liquids in deformable porous media from the momentum equation of the liquid constituent

V( - v,y = - F(VP( -p(g), (A%

where K is the permeability tensor. [E, is the Lagrangian strain tensor, ~1 is the matrix viscosity, and P& is the pore liquid pressure (see, for instance, Ref.“, or Ref.14 which focuses on the application to composite materials manufacturing.)

Manipulating the model as done in Ref.“’ one can then write

g + V.(&v,) = 0

v.v, = +ww, - Pcg)l (A7)

wP.( ~+v,3v,) = V.T,,, + h?4 S.Pf( 1 - 4,)ls (A8)

In this model the only quantity which is still constitutively to be specified is the stress tensor for the mixture as a whole T,. Information on it, however, can be obtained using classical continuum mechanics tools and validation can, in principle, be performed, directly by conceiving suitable but standard experiments, aimed at evaluating the dynamical response of the wet solid preform compressed at different

17


volume ratios to oscillatory twist and compression, and to creep and stress relaxation tests.

Unfortunately, the literature lacks of experimental results in this direction. Only Kim et al. 4 performed experiments addressed to the study of the stress relaxation properties of a wet preform finding a spectrum of relaxation times, and several other authors 2,10,1 l-16 have noticed a viscoelastic behaviour of the materials they use with non-negligible relaxation times.

On the basis of these (mostly qualitative) observations, it has been suggested 2o that the wet preform be modelled as an anelastic solid or as a Voigt-Kelvin solid and experiments in this direction are encouraged. However, at present it is hard to identify the values of the physical parameters involved. In this situation, here it is assumed that both the wet and the dry preform respond elastically to deformations.

It is also useful to observe that adding the two continuity eqns (Al) and (A2) one has that the so-called composite velocity

v, = W, + We, (A9)

is divergence-free

VT, = 0. (Al@

In one-dimensional problems, determining explicitly ve from Darcy’s law (eqn (A5)) and exploiting the fact that eqn (AlO) implies that the composite velocity is constant, one can then deduce the following model

(All)

where 7: is the excess stress of the solid-liquid mixture. If a material surface of discontinuity u is present within

the domain, then, following Liu 24 and Mtiller 25, one can deduce for eqns (66)-(68) the following set of interface conditions

[[v&t = 0, (A13)

IF&r - W#Jsv,(v, - v&D = 0, (A14)

where n is the normal to the surface of discontinuity, v, its velocity, and u] denotes the jump off across it. These conditions represent continuity of the fluxes through the interface, that is conservation of mass and momentum.

According to eqn (A13), the composite velocity is continuous across each material interface. Since, as already stated, in one-dimensional problems it is constant within each domain, then it is constant everywhere. In particular, going all the way down where there is only pure liquid, one has that the composite velocity is everywhere equal to the inflow velocity

v,-(x, t) = uin(t) VX E D, (A13

and therefore,

v&9 t> = 4nCt> - +s(-G t)vs(x9 t,

1 - b,(x, t> . 6416)

REFERENCES

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Modelling matrix injection through elastic porous preforms

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