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Modeling of fiber orientation in fiber filled thermoplastics 29-03-2010. Prof.dr.ir. D.J. Rixen Prof.dr.ir. L.J. Sluys Dr.ir. M.J.B.M. Pourquie. Adriaan Sillem TU Delft, numerical analysis / solid and fluid mechanics / construction mechanics DSM, materials science center. - PowerPoint PPT Presentation
Modeling of fiber orientation in fiber filled thermoplastics 29-03-2010Adriaan SillemTU Delft, numerical analysis / solid and fluid mechanics / construction mechanicsDSM, materials science centerProf.dr.ir. D.J. RixenProf.dr.ir. L.J. SluysDr.ir. M.J.B.M. Pourquie
Prof.dr. J.J.M. SlotDr.ir. H. van Melick
Introduction.Application.Models.Test cases.Conclusions and recommendations
LayoutINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
IntroductionINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Master student mechanical engineering.Final project of mechanical engineering study.Carried out at DSM, a Dutch multinational Life Sciences and Materials Sciences company.
ProjectINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
DSM is interested in simulating the production process injection molding.
InterestINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Implementation of a model to gain insight in the fiber orientation development during injection molding.
ObjectiveINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Reproduction of literature material.Implementation of models.Interpretation of output.Comparison of quality of approximations.Instruction of implementation.Integration of all of the former in a report.
ContributionINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
ApplicationINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Focus of the project is on thermoplastic products.
In our daily lives, we encounter thermoplastic products.Thermoplastic productsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
ExamplesThermoplastic productsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Used materials in granules or pellets form.MaterialINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Injection moldingINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Used materials in granules or pellets form.MaterialINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Add discontinuous fibers to improve the quality of the final product. Glass or carbon fibers, for instance.
Influences:viscosity,stiffness,thermal conductivity andelectrical conductivity.
DSM wants to controlthese properties.
MaterialINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Typical volume fractions are 8-30%.Typical weight fractions 15-60%.Typical average final lengths 200-300 [m].Typical diameter 10 [m].
Average human hair is100 [m].
MaterialINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Configuration of the fibers.
Configuration of the fibersINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
How is the configuration of the fibers controlled?
We want to control the production process.Configuration of the fibersINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Configuration of the fibers can be controlled byprocessing settings on the injection molding machine andmold geometry (expensive!).
Simulations are preferred.
ControlINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
DSM already uses the commercial softwareMoldflow andMoldex3Dto simulate injection molding processes.
Problems encountered by DSM:results warpage (kromtrekking) analyses not good,results fiber orientation analyses not good,company secrecy response concerning the code,models not up to date.
SoftwareINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
DSM wants implementations of the latest fiber orientation models.
ProjectINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
How can the configuration of the fibers be described?Location of the fibers.Orientation of the fibers.
It is reasonable to assume that the concentration of the fibers in the final product is uniform by approximation.
The orientation of the fibers influences the material properties the most.
We will focus on the modeling of the orientation.Fiber orientationINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
ModelsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
What are the physics behind the fiber orientation? What are the characteristics?Inertia forces are negligible low Reynolds number.Viscous forces lift and drag.Stiffness of the fibers internal stresses.Fiber-fiber interaction fibers hit each other.Fibers influence surrounding flow field coupling of orientation and velocity.
Order of increasing complexity of the models.Physics behind fiber orientationINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Models are constructed for incompressible homogeneous flows. Such flows are characterized by velocities whereof the change in space is constant.
The velocity can be described as
Incompressibility demands thatIncompressible homogeneous flow fieldsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Examples arethe simple shear flow field
the uniaxial elongational flow fieldIncompressible homogeneous flow fieldsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Layout modelsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Jefferys equationEllipsoidal particle immersed in a viscosity dominated fluidINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Jefferys equationINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Jefferys equationINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Balance between viscous stresses from the flow field and internal stresses of the fiber leads to
The re is the aspect ratio of the ellipsoidal particle. The is a function of this aspect ratio.Jefferys equationINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Layout modelsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Model fiber-fiber interaction deterministically or stochastically?Folgar-Tucker modelINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Outcome of a throw of a dice.Hard to say in a deterministic sense.Easy to say in a stochastic (probabilistic) sense.
Same holds for fiber-fiber interaction.Folgar-Tucker modelINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Folgar-Tucker modelThree equations, four unknowns, not solvable. INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Folgar-Tucker modelINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Folgar-Tucker modelFour equations, four unknowns, so solvable. INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Folgar-Tucker model
Viscous and internal stresses of a convective nature.Fiber-fiber interaction of a diffusive nature.INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Jefferys equationINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Partial information
Solving the probability density is very demanding in the computational sense.
Is there a computationally less expensive approach?
INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Layout modelsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Partial information
--> less information
The only unknowns are the orientation tensors
Sum is infinite! Maybe a set of lower order terms already give enough information on the fiber orientation.
INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Partial informationLet us first consider the symmetric second order tensor.
p1p1 sin p2p2 sin p3p3 sin p1p2 sin p1p3 sin p2p3 sin INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Partial information
The orientation tensor already gives sufficient information on the fiber orientation.
But can it be solved more efficiently?INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Folgar-Tucker tensor form
The rate equation for A.
Not a function of orientation p. Only of time t.
We have realized a reduction of the #DOF from three to one. The tensor form can be solved more efficiently.
INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Folgar-Tucker tensor form
The rate equation for A.
Comparison with experimental results show that the transient time is too short.
INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Layout modelsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Correction
We need to correct the rate equation.
How should we do this?INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
CorrectionA straightforward way would be
Model is not objective.INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Wang-OGara-TuckerIn 2008, Wang, OGara and Tucker found a way to prolong the transient time in an objective way.
Because the orientation tensor is symmetric, it has a spectral decomposition.INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Wang-OGara-TuckerIs objective!
But the equation is not solvable.INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Closure approximationFourth order tensor is unknown.
With relations with the second order orientation tensor and solving for some easy cases, we can construct
and solve the rate equation.
F is called a closure approximation as it close the problem in an approximate way.INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Closure approximationThe idea is fitting.
The F can be constructed in many ways!INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Closure approximation
We have to investigate the quality of the approximation.INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Layout modelsDone!INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Solution strategyWe solve the equations withfinite volume method in space andEuler forward in timefor the probability density function, andEuler forward in timeseveral closure approximationsfor the orientation tensor.
We have to compare the quality of the approximations.INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Test caseINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Simple shear flow field is described by
v = velocity in [m/s]x = space in [m]G = shear rate in [1/s]Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Simple shear flow fieldSimple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Partial informationLet us consider the symmetric second order tensor.
p1p1 sin p2p2 sin p3p3 sin p1p2 sin p1p3 sin p2p3 sin INTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Balance between contributions can be varied by changingthe initial condition andthe time dependence of the strain and shear rates.
Gives an indication of dependence on history.Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
ICs taken with orientation already towards the x1 directionSimple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Phenomenological parameter kappa to prolong the transient time.Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Comparison of the solutions of the kinetic theory and the tensor approximations.Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Taking another parameter set results in a decay of the quality of the approximations.Simple shearINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Conclusions and recommendationsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
ConclusionsThe numerical approach for the probability density and orientation tensor are sufficiently fast.The convection and diffusion of the probability density strongly depend on the initial condition and the time dependence of the strain and shear rates.The slow down will help fitting to experimental results.The quality of the closure approximations varies with the parameters and the flow fields.
RecommendationsFurther study of the literature.Closure approximations should be investigated.Conclusions and recommendationsINTRODUCTION | APPLICATION | MODELS | TEST CASES | CONCLUSIONS AND RECOMMENDATIONS
Questions?
Good afternoon everybody. My name is Adriaan Sillem.
I have the honor to give a presentation about the work that I have been doing the previous 6 months.
It is about the modeling of fiber orientation in fiber filled thermoplastics.
Difficult title, it will become apparent in a moment.*This is the layout of my presentation.
I will start with an introduction,
followed by an explanation of the application in question.
Afterwards, I will treat some models that will simulate the application.
I will treat an example, a test case,
and I will end the presentation with my conclusions and recommendations.
**and Materials sciences company, and it is the case that. (next slide)*. DSM is interested in simulating the production process injection molding.*In accordance, the objective of the project was to implement a model to gain insight in the fiber orientation development during injection molding.
Again, I emphasize that the difficult terms will become apparent in a moment.*My contribution to DSM was
the reproduction of literature material on the models,the implementation of the models,interpretation of output for certain test cases,the comparison of the quality of several approximations anda few weeks ago, I gave an instruction at DSM on the usage of my programs, so that DSM personnel knows how to use them.I integrated all of the former material in a report so that a complete package of all my work is available.
Thats it for the introduction, I will continue with the application.*I am now done with the introduction and tell something about the application in question.***All of the former thermoplastic products were originally in granules or pellets form.*I will now show a movie on how injection molding works.
****Say we would, hypothetically, take a finalized thermoplastic product and burn away the matrix material, the material that surrounds the fibers, then all that is left are the fibers.
The fiber locations and orientations vary throughout the product.
**Examples of processing settings on the injection molding machine areinjection speed andpressure.
Examples of the mold geometry areinjection location andmold cavity, correcting the shape if the final product does not have the desired shape.
Correcting the mold geometry is very expensive,
so simulations of the process are preferred,
as they can be performed relatively cheaply.*In accordance, DSM has some commercial software in place, namely Moldflow and Moldex3D.
However, DSM encountered some problems, namely that.
The theoretical results do not resemble the experimental results.
Warpage and fiber orientation are closely related.
Company secrecy response makes it harder to solve problems encountered in the simulations.
It can take quite a while before new models from academia are implemented.
Due to these problems, this is where.(next slide)*.my project comes in, because
DSM wants implementations of the latest fiber orientation models.
Get a feeling in how the models work and what the influences of the parameters are.Then DSM will have more insight in the origin of modeling problems, andcan discuss with the software industry.
DSM is, however, not interested in building a complete injection molding software package or to compete with the other commercial software companies.
Just knowledge on these lower levels.
In doing so, we have to ask ourselves (next slide)
*. how can the configuration of the fibers be described?
Location of fibers.Orientation of the fibers.
As a first assumption to construct a model for the configuration of the fibers, it is reasonable to assume that .*We understand the application now, so how are we going to construct a model?*Molten thermoplastic material flow has a low viscosity.When determining the ratio between the inertia and viscous forces, we find that it is low. For the engineers among us, the Reynolds number is low.Thus inertia forces can be neglected.
Rest of story is evident.*First I must say that the models were all constructed for incompressible homogeneous flows.
Mind that K and c can still be functions of time t.*As you can see, K is constant in space and the trace of K is zero.
I will depict the upper flow field later, for now it is just to give two examples.*The purpose is to implement a model from 2008.
In doing so, it is didactical to treat some preliminary models.
Thus I will start with a simpler model, called Jefferys equation.
We will progress in increasing order of complexity.
I will show this scheme more often throughout this presentation to indicate how we are progressing.*The latter is also called a Stokes flow.*Balance between the viscous stresses from the flow field and internal stresses of the fiber.
Length of the long axis of the ellipsoidal particle is denoted with l, and the direction with the unit vector p.*One end point always in the origin, other always at distance one from the origin, thus on the surface of the unit sphere, which we will denote with capital S.
The p can be described by two DOF, theta and phi, also called the elevation and azimuth.*Resulting formula is.
Dont expect you to know it by heart, it is just to give an idea on how the equations look like.*Jefferys equation did not take FFI into account.
The next model, the Folgar-Tucker model does.
We are still constructing a model to simulate fiber orientation! Nothing more.*I will explain this problem with (next slide)* an example of the throw of a dice.
Fiber-fiber interaction is modeled as a random force on each fiber, being hit by other fibers,
which changes. (next slide)*Jefferys equation with only viscous and internal stresses to the Folgar-Tucker model that also takes FFI into account.
I will skip the derivation of the FFI term.
What is important is that it depends on factors like
viscosity, in C_I,strength of the flow field, in gamma_dot andvolume fraction of fibers in thermoplastic material, in psi.
It is important to remember what psi means. It is the probability density of the fiber orientation p, thus also lives on S.
When it is large, it is expected that the fiber is oriented in direction p, when it is small , it is not expected that the fiber lies in direction p.
If psi is known, we have an equation for p. But psi is unknown, so we have three equations and four unknowns. We are short one equation!
Well have to find another equation.*. which comes in the form of a conservation equation for probability.
Say we consider an ARBITRARY element S on S.
Then the change of the probability density, is
what comes in, minus
what goes out!
If we write this in mathematical form we get. (next slide)*this equation.
Now four equations and four unknowns, which is solvable! *Rewriting the equation, we get this form. We see that
the first term results in convection of probability density, the last term results in diffusion of probability density.*Viscous and internal stresses in convection. Motivated by.
And FFI in diffusion, motivated by
*It unfortunately turns out that the probability density is computationally expensive to solve.
*Lets treat the orientation tensor approximation.
We are still building a model for the fiber orientation, but now we hope to construct a fast one!*Onat and Leckie constructed this expansion.
Sort of Fourier expansion.
To the right, contains less and less information. Amount of information converges to zero.
The only unknowns are the orientation tensors.
Sum is infinite, maybe a few lower orders terms give enough information.*Let us consider the first unknown orientation tensor.
The psi is weighted in a integral over S.**A rate equation can be deduced for the second order tensor A.
We do not have to solve in orientation p anymore, only in time t!
So more efficient!
But there is still a disadvantage. (next slide)*. comparison with experimental results shows that the fibers align to fast. In other words, the solution converges to fast to its steady state or final state.
How do we correct this?
*Leads to the final models!
We can simulate the fiber orientation,
but hope to correct the time scale in which the process is completed.**Explain objectivity very simple.**Everything is known except for the fourth order tensor.*F is called a closure approximation.*The idea is fitting a curve to known data points.
Explain fitting a bit.
We approximate the solution.
Bare in mind that fitting can be done in many ways!
So there are many possible approximations.**We have found a relatively cheap way to get information on the fiber orientation!
We are done with the models and can now test them,
if we can find a solution.*We find a solution with the following strategy.**As an example I chose the simple shear flow field.
G is the shear rate.*Particles move like this.. (show a bit with hands).
What do we expect for the orientation?
To keep tumbling if they have a thickness.
If they are very thin, they will align.
We will concentrate on the latter!*Explain results a bit.**Components of A.
Recall the interpretation a bit.
Explain why the progression is like this
Notice the overshoot. Balance between convection and diffusion.
A12 is not zero because FFI pulls to random fiber orientation.***Change in balance due to different ICs.
I can change the balance between the viscous and internal stress and the fiber-fiber interaction.
Note the dependence on the history.*Time dependent shear rate.*Time dependent shear rate.
I can change the balance between the viscous and internal stress and the fiber-fiber interaction.**Approximating the exact solution of A.
We want a cheap solution! So find a good approximation.
Results are good for the black line, the OFC.
Closure was taken for a parameter set, that was constructed with this particular flow field and parameters.*Changing the parameter set leads to poor results.
Can be explained by the fact that the convection to diffusion ratio is greater than in the previous case. The contribution of the closure is thus stronger.
Other cases have to be investigated to ensure a good overall quality of the approximation.
DSM has to been provided with the codes and can now do this research.*I have explained the application,
constructed the final model,
treated an example.
From this, we can draw some conclusions and make some recommendations.***