Inverse Shape Design in Injection Molding Based on the Finite Element Method

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Inverse Shape Design in Injection Molding Based on the Finite

Element Method

Inverses Design in Spritzgussverfahren basierend auf der Finite-Elemente-Methode

Von der Fakultät für Maschinenwesen der Rheinisch-Westfälischen Technischen Hochschule Aachen zur Erlangung des akademischen Grades eines Doktors der Ingenieurwissenschaften

genehmigte Dissertation vorgelegt von Florian Zwicke

Berichter: Univ.-Prof. Marek Behr, Ph. D. Prof. Karen Veroy-Grepl, Ph. D. Tag der mündlichen Prüfung: 12.05.2020

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Abstract

A method is proposed for the inverse design of cavity shapes for the injection molding pro-cess. When liquid polymer melt is cooled down in an injection mold to manufacture plastics parts, inhomogeneities in the cooling and solidification processes lead to shape defects in the finished molding. The geometry of the cavity where the liquid melt is injected is largely re-sponsible for the shape of the molding. The method described in this document offers an automatized tool for the determination of a suitable cavity shape that will reduce faults in the molding shape.

The basis of this method is a numerical simulation of the injection molding process. This method builds on simulation models for both fluid and solid polymers that incorporate the important physical phenomena of thermoviscoelastic material behavior and solidification. Separate simulation models are described in this document for the solidification and post-ejection stages of the process. They are both equipped with a finite element formulation that makes them suitable for a swift implementation in a computer code.

The inverse design method for the cavity shape results from a combination of an inverse for-mulation of stationary thermoelasticity with an iteration scheme that incorporates the non-elastic effects. This iterative method is demonstrated for two sample cases. The simulation method is shown to represent the important aspects of the viscoelastic behavior and solidifica-tion. The iterative inverse design method produces suitable cavity shapes after small numbers of iteration steps. Furthermore, plots of a distance measure over the course of the iteration indicate rapid convergence of the method.

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Zusammenfassung

Es wird eine Methode für die inverse Auslegung von Kavitätsformen für Spritzgussverfahren beschrieben. Wenn zur Fertigung von Kunststoffbauteilen flüssige Polymerschmelze in der Kavität abgekühlt wird, dann kommt es durch Ungleichmäßigkeiten in den Abkühl-und Erstarrungsprozessen zu Formdefekten im fertigen Bauteil. Für die Form des Bauteils ist hauptsächlich die Form der Kavität verantwortlich, in welche die flüssige Schmelze eingespritzt wird. Die Methode, welche in diesem Dokument beschrieben wird, stellt ein automatisiertes Werkzeug zur Bestimmung einer geeigneten Kavitätsform bereit, welches Fehler in der Bauteilform reduzieren kann.

Die Grundlage dieser Methode ist eine numerische Simulation des Spritzgussprozesses. Die Methode baut auf Simulationsmodellen für sowohl flüssige als auch feste Polymere auf, welche die wichtigen physikalischen Phänomene des thermoviskoelastischen Materialverhaltens und der Erstarrung berücksichtigen. In diesem Dokument werden für die Prozessphasen während der Erstarrung und nach dem Auswurf unterschiedliche Simulationsmodelle beschrieben. Diese sind beide mit Finite-Elemente-Formulierungen ausgestattet, wodurch sie sich mit wenig Aufwand in einen Programmcode implementieren lassen.

Die inverse Methode zur Gestaltung der Kavitätsform entsteht aus der Kombination einer inversen Formulierung stationärer Thermoelastizität mit einem Iterationsschema zur Berücksichtigung nicht-elastischer Effekte. Die iterative Methode wird an zwei Beispielen demonstriert. Es wird dabei gezeigt, wie die Simulationsmethode die wichtigen Aspekte des viskoelatischen Materialverhaltens und der Erstarrung abbildet. Die iterative inverse Methode erzeugt schon nach wenigen Iterationsschritten geeignete Kavitätsformen. Darüberhinaus legen Plots eines Ähnlichkeitsmaßes über den Verlauf der Iteration eine rasche Konvergenz der Methode nah.

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Acknowledgments

I have carried out the work on my research project and this dissertation at the Chair for Com-putational Analysis of Technical Systems (CATS) at RWTH Aachen University. Throughout the years I have received a lot of help and support from my colleagues and I am grateful to all of them. There have been countless interesting scientific discussions, but at least as impor-tant were the social and recreational activities organized by various colleagues that helped take our minds off of work. In the following, I will mention certain people who have made a particular contribution to my work or my dissertation.

I wish to thank my supervisor, Prof. Stefanie Elgeti, who has helped me tremendously throughout the years. Her contributions did not only include scientific advice on the direction of my research, but she has helped me deal with the various tasks involved in setting foot in the academic world. This includes paving the way for various international conference visits, as well as a research stay at the University of Texas at Austin.

Furthermore, I am grateful to Prof. Marek Behr for giving me the opportunity to be a part of CATS and to work on this interesting research project, and also for reviewing my thesis. Thanks go out, as well, to Prof. Karen Veroy-Grepl for agreeing to be the second reviewer on my thesis.

While working on my project, I have had particularly helpful and frequent scientific dis-cussions with a number of colleagues and friends. I wish to express my gratitude to Philipp Knechtges and Roland Siegbert, who have always been ready to share their expertise on topics such as physics or computer science, and have both played pivotal roles in shap-ing my early years at CATS. I am very grateful to Stefanie Günther, who I have frequently turned to for discussions on inverse problems and related mathematical theory. Thanks go out to Thomas Ludescher, for all the helpful discussions and advice throughout the years on many theoretical aspects of my work. And, finally, I thank Sebastian Eusterholz for sharing his expertise, particularly in mechanical engineering, on many occasions.

During my stay at CATS, I have had the opportunity to advise a few great students. They have supported me either as student assistants or by writing student theses, and have also taught me a lot of things. I wish to thank Marko Blatzheim, Konstantin Key, Mario Koretz, Saša Lukić, Eugen Salzmann, and Daniel Wolff for their great contributions to my project and work.

The months that I spent working on my dissertation and preparing my doctoral exam were a very stressful time. I am particulary grateful to the people who have helped and supported me in this time. For their important and detailed feedback on my dissertation, my gratitude

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belongs to Philipp Knechtges and my father. For the help I received in preparing and orga-nizing my doctoral exam, I wish to particularly thank Michel Make. I am further grateful to Stefanie Günther, Philipp Knechtges, and Prof. Stefanie Elgeti for taking the time to give me such helpful feedback on my defense talk.

Not all the support that I have received in this time was of a strictly scientific nature. However, it was certainly no less important. I am deeply grateful to Moritz Begall and Louisa Ludwig-Begall for their support in this sometimes challenging time. Last, but not least, I wish to sincerely thank my parents, my sister, and my two young nieces, who did not get to see me very often in this time, for all their patience and support.

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List of Figures

1.1 The shape of a molding that is produced by injection molding differs from the cavity shape in the mold. These shape differences result from the cooling-induced material shrinkage and residual stresses caused by the solidification. . 1 1.2 The main objective of this document is to develop a method that can predict

a cavity shape for a desired molding shape. . . 2 2.1 An elastic spring with stress 𝜏 and strain (i.e., relative length change) 𝛾. . . 6 2.2 The strain 𝛾 is shown if a constant stress ̄𝜏 = 1 is suddenly applied at 𝑡 = 0 to

a Hookean solid with modulus 𝜇 = 1. . . 6 2.3 A viscous dashpot with stress 𝜏 and strain 𝛾. . . 7 2.4 The strain 𝛾 is shown if a constant stress ̄𝜏 = 1 is suddenly applied at 𝑡 = 0 to

a Newtonian fluid with viscosity 𝜂 = 1. . . 7 2.5 The Maxwell fluid model is comparable to an elastic spring and viscous

dash-pot combined in series. . . 8 2.6 The strain 𝛾 caused in a Maxwell fluid by a stress ̄𝜏 = 1 that is suddenly applied

at 𝑡 = 0 is shown. The viscosity is chosen as 𝜂 = 100 and the modulus as 𝜇 = 1. The plot is shown for a shorter (left) and a longer (right) observation time. Hookean solid and Newtonian fluid behavior are shown for comparison. In the beginning the curve is similar to that of a Hookean solid, but tends to that of a Newtonian fluid in the long term. . . 8 2.7 Spring-dashpot combinations as in the Maxwell model are combined in

par-allel to model multiple relaxation times. . . 10 2.8 A mechanical model that is similar to the Oldroyd-B material is obtained

by adding a dashpot in parallel to the spring-dashpot combination from the Maxwell model. . . 10 2.9 The strain 𝛾 is plotted over time when a stress ̄𝜏 = 1 is suddenly applied to

an Oldroyd-B fluid at 𝑡 = 0. The viscosities are set to 𝜂 = 1 and 𝜂′ _{= 99, and} the modulus to 𝜇 = 1. Newtonian, Hookean, and Maxwellian materials are shown for comparison. At first, the material behaves viscously (left), later it shows similarities to elastic behavior (middle), before approaching fluid be-havior again in the long term (right). . . 11 2.10 Strain curves for Maxwell (𝜂 = 100, 𝜇 = 1) and Oldroyd-B (𝜂 = 1, 𝜂′ _{= 99,}

𝜇′ _{= 1) fluids are shown when a stress ̄𝜏 = 1 is suddenly applied at 𝑡 = 0.} Due to the logarithmic plot, all the different behavioral regimes are visible, including the difference in behavior at high frequencies. There is a clearly visible plateau — the rubber plateau — in the Oldroyd-B curve. . . 12

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2.11 The Kelvin-Voigt solid can be compared to a mechnical model with an elastic spring and a viscous dashpot in parallel. . . 13 2.12 The strain 𝛾 is shown if a stress ̄𝜏 = 1 is suddenly applied to a Kelvin-Voigt

solid at 𝑡 = 0. The viscosity is chosen as 𝜂 = 1 and the modulus as 𝜇 = 1. Both short-term behavior (left) and longer-term behavior (right) are shown, in comparison to Newtonian, Hookean, and Oldroyd-B materials. The material at first reacts like a viscous fluid, but then tends to elastic material behavior in the long term. . . 13 2.13 The strain 𝛾 in Kelvin-Voigt (𝜂 = 1, 𝜇 = 1) and Oldroyd-B (𝜂 = 1, 𝜂′ _{= 99,}

𝜇′ _{= 1) materials is compared when a stress ̄𝜏 = 1 is suddenly applied at} 𝑡 = 0. All different behavior regimes are visible in the logarithmic plot. The materials’ curves are similar in the beginning and leading up to the rubber plateau. Afterwards, the Kelvin-Voigt curve sticks to the Hooke curve, while the Oldroyd-B curve approaches a Newtonian curve. . . 14 2.14 The standard linear solid model can be compared to a mechanical model with

an elastic spring in parallel to the spring-dashpot combination we know from the Maxwell model. . . 14 2.15 This is a depiction of a sparsity pattern of a matrix that was obtained in an

iteration step of Newton’s method in the solution procedure of finite element-discretized equations for viscoelastic fluid flow. . . 17 3.1 The coordinate system maps ̃𝝌A and ̃𝝌B map from the reference coordinate

system to coordinate systems that describe different material states. The co-ordinate transformation map [𝝋B

A] is the counterpart of this map between two material states which are different from the reference state. . . 21 3.2 There are five different material states that are relevant for this stationary

thermoelasticity formulation. These are grouped into virtual and complete material states. The original and stress-free states are related by thermal ex-pansion or contraction. The initial deformation defines the initial state. Two displacement fields describe the deformation in the initial and physical states with respect to the reference state. The reference state is where the differen-tial equations are formulated and serves as the simulation mesh in a numerical setting. Different types of deformation measures are available that describe the relationships between these material states. The tensor [𝐁P

Z] describes the total deformation in the physical state with respect to a stress-free state. This can be calculated from the other deformation measures, as depicted at the bottom of the figure. . . 23 3.3 A regular Lagrangian formulation is obtained if initial and reference state are

chosen to be identical. In this case, the equations are formulated in the co-ordinate system that corresponds to the initial state. These equations can be solved for the unknown physical state. A mesh that is used in a numerical simulation would correspond to the initial state. . . 24

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3.4 An Eulerian formulation is obtained if physical and reference state are chosen to be identical. In this case, the equations are formulated in the coordinate system that corresponds to the physical state. They can be solved for the unknown initial state. A mesh that is used in a numerical simulation would correspond to the physical state. . . 24 3.5 The stress-free material state results from an unconstrained deformation of

the original material state due to variations in temperature. We assume isotropic thermal expansion, such that the determinant of deformation [𝐽Z

O] is a sufficient measure for describing this deformation. . . 25 3.6 The existing deformation in the initial state is prescribed by means of the left

Cauchy-Green deformation tensor [𝐁I

O]. Depending on how this initial defor-mation was obtained, the original state may not have an associated coordinate system. . . 26 3.7 The physical and initial material states are both described by displacement

fields with respect to the reference state. The deformation gradient tensors can be calculated directly from the displacement field gradients. . . 26 4.1 We use a rough mesh for this simulation to better show the effect of the inverse

method on the output mesh. The shape can be interpreted as a bridge that is embedded in the ground. . . 39 4.2 The boundary conditions of a forward run of the simulation are sketched in

this figure. Dirichlet boundary conditions are defined on the entire boundary for the initial displacement field (seen on the left-hand side), to set initial and reference states to be identical. On the lower boundary parts, Dirichlet bound-ary conditions are defined for the displacement field to embed the body in the ground. Neumann boundary conditions for zero boundary forces are imposed on the remainder of the boundary, to allow it to move freely and take on its equilibrium shape. . . 39 4.3 The simulation mesh that corresponds to the initial state is identical to the

one in the reference state. . . 40 4.4 In the simulation mesh that corresponds to the physical material state, the

deformation of the body due to gravity is clearly visible. The center part of the body bent downwards, while the two ’legs’ of the body are slightly bent outwards. . . 40 4.5 The one-dimensional stress is shown in the initial (left) and physical (right)

states, i.e., without forces applied and with gravity. Since there is no initial deformation, the stress is zero if no forces are applied. When the body is deformed due to gravity, stress peaks are visible in the inside corners. . . 40 4.6 The boundary conditions for the inverse run of the simulation are shown in

this sketch. The displacement field for the shape in the physical state is now prescribed on the entire boundary to fix the shape that is reached when grav-itational forces act. Double Dirichlet boundary conditions for both displace-ment fields are prescribed on the lower boundaries since their positions are fixed and arbitrary forces are allowed. The remainder of the boundary still needs Neumann boundary conditions since zero boundary forces act there. . . 41

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4.7 The simulation mesh that describes the body’s shape in the initial state in the inverse simulation is shown in this figure. The body is bent upwards in the middle, and slightly inwards on the sides. . . 41 4.8 The body’s shape in the physical state in the inverse simulation is shown.

When gravitational forces act on this body, it deforms to take on the desired shape. . . 41 4.9 The one-dimensional stress in the body is shown for the inverse variant of

the simulation. There is no stress if no forces act on the body. When it is deformed due to gravity, there is a similar stress field as in the forward simu-lation. This shows that the inverse method, while achieving the desired shape exactly under deformation, does not remove residual stress. . . 41 4.10 Simulation mesh and desired shape for the thermoelasticity simulation. . . 42 4.11 The simulation mesh in the initial state is identical to that in the reference

state when the simulation is run in forward mode. . . 43 4.12 The mesh in the physical state is severely deformed due to thermal contraction

in the body’s interior. . . 43 4.13 The initial temperature in the initial state (left) and the temperature in the

physical state (right) of the body are shown. The initial temperature field is the result of the heat equation simulation. Since the material was cooled down from the outside, hot parts remain in the interior. The body is fully cooled down to the uniform boundary temperature in the physical state. . . 44 4.14 One-dimensional initial (left) and physical (right) stress in the body. Since

initial deformation is not used, there is no stress in the initial state. Deviatoric stress builds up during the cooling such that residual stresses remain in the physical state. . . 44 4.15 Shape of the body that results in the initial state in inverse mode. The body

is much larger than the original simulation mesh to make up for the thermal contraction. . . 45 4.16 The mesh that describes the shape of the body in the physical state in inverse

mode is not identical to the simulation mesh, except on the boundary. This is because deformation in the interior is still allowed to avoid distortions of the initial temperature field. . . 45 4.17 This figure shows the initial temperature in the initial state (left) and the

tem-perature in the physical state (right) that are calculated in the inverse simula-tion. The initial temperature field is transformed by the initial displacement field that is determined by the smoothing equation. This is just an estimation of how the initial temperature field could look if the heat equation was solved on the changed geometry. . . 45 4.18 The initial (left) and physical (right) stress is shown in the figure. Residual

stress remains in the physical material state, since the inverse method is de-signed specifically to yield the perfect physical shape, but does not reduce stress. The stress that is calculated in the simulation can help estimate whether the material will undergo plastic deformation or break. . . 45

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5.1 Three different types of material states are considered in the Eulerian frame. The equations are formulated in the physical state, which also corresponds to the simulation mesh in a numerical simulation. The density change describes the volumetric deformation of the physical state with respect to the reference state. The thermally deformed state is the material state at the reference pres-sure and the physical temperature. . . 51 5.2 The conformation tensor, with any nonzero relaxation term, can be

under-stood to describe the deformation of the physical state with respect to two additional material states, ̄R and ̄Θ. The relevant deformation measures that describe these states are similar to those that describe the reference and ther-mally deformed states, R and Θ. . . 57 5.3 The thermal expansion factor 𝛾 defines the deformation between the reference

and thermally deformed states. The determinant of deformation between the physical and thermally deformed states can be calculated directly from the thermal expansion factor and the density ratio. . . 63 5.4 In a theoretical fully relaxed state, the physical and thermally deformed states,

P and ̄Θ, are identical. This means that the conformation tensor 𝐊 describes the deformation due to unconstrained thermal expansion or contraction. . . . 71 5.5 The simulation mesh that is used as the closed cavity shape. . . 78 5.6 The relationships of temperature 𝜃, the phase parameter 𝜙 and the effective

relaxation time 𝜆effare shown in these plots for the parameters chosen in this testcase. The interace temperature ̄𝜃 = 320, as well as the phase parameter value 𝜙 = 0.5 and the timestep and simulation lengths are shown in the plots. The effective relaxation time is much smaller than the timestep length at the initial temperature 𝜃 = 350, but increases to a value larger than the full sim-ulation time at the temperature 𝜃 = 300. This means that material at this temperature will appear solid in this simulation. . . 80 5.7 The thermal expansion factor 𝛾 is plotted over temperature 𝜃 for the

exponen-tial thermal expansion law with 𝛼Θ = 0.005. The maximum volume change under these conditions is around 22%. . . 80 5.8 Velocity, density, and temperature fields are shown at different points in time

during the cooling process. The velocity direction is indicated by arrows and the phase interface is indicated by a line at 𝜙 = 0.5. At first, when most of the material is still fluid, we can see how the material density increases due to cool-ing and there is flow from the interior towards the boundaries. The density gradient in the interior grows as more material shrinks near the boundaries. Eventually, the flow field vanishes, as the material solidifies, and the density gradient decreases to a small nonzero value. . . 81 5.9 The vertical component of velocity is plotted over time together with the

phase parameter. There is still periodic motion in the material after solidi-fication is complete, because it behaves elastically. This motion is dampened due to the remaining viscous stress contribution. . . 82

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5.10 The average density and density range are shown in this plot. Since the cavity is closed, mass is conserved, and the average density stays fixed. There are initial significant spatial variations in the density. They decrease over time, but due to solidification and the build-up of shear stress, they do not completely disappear. . . 82 5.11 Both average temperature and temperature range are plotted over time. The

temperature reaches a value very close to 𝜃 = 300 everywhere during the runtime of the simulation. The interface temperature 𝜃 = 320 is reached very early. . . 82 5.12 The phase parameter field is shown at different points in time. The variable

thickness of the phase interface is visible in these pictures. . . 82 5.13 The scalar stress measure is shown at different points in time. The deviatoric

stress slowly builds when the material is solid. The motion caused by the remaining density gradient leads to the build-up of deviatoric, or shear, stress, with a few peaks in the inside corners. . . 83 5.14 The average deviatoric stress is plotted over time. There is a fast build-up of

stress as the material starts cooling down. At first this stress, also relaxes quite quickly. Eventually, this relaxation becomes very slow and residual stress re-mains in the material. . . 83 5.15 Pressure and determinant of deformation (i.e., the volumetric part of

defor-mation) are shown in this plot. The pressure decreases considerably as the material cools down but is kept from shrinking. . . 83 5.16 Simulation mesh for the open cavity. . . 84 5.17 Boundary conditions for the solidification simulation. At the inlet on the

left-hand side, material is allowed to flow in at a constant pressure 𝑝0 = 0 and heat transfer is purely advective. At the cavity walls, the material sticks to the boundaries and a constant temperature is prescribed. . . 84 5.18 The relationship of phase parameter, temperature, and effective relaxation

time is shown. Special lines are drawn at the phase interface at 𝜙 = 0.5 and 𝜃 = 325, and also at the time step and simulation length. The relaxation time is far below the time step length at high temperatures, but far above the sim-ulation length at low temperatures. This means that the material behavior, as it is observed in the simulation, switches from mostly fluid to mostly solid behavior. . . 86 5.19 The velocity field is shown at different points in time. In the beginning,

ma-terial flows towards the boundaries due to the sudden temperature-induced shrinkage at the walls. Material then starts to flow in at the inlet until the ma-terial has solidifed too much. There is still motion after solidification, which contributes to the build-up of shear stress. . . 86 5.20 The temperature field is shown at various points in time. The phase

inter-face at 𝜃 = 325 is drawn as a black line. Due to the influx of hot material, the temperature near the inlet stays high longer than in the remainder of the material. . . 87

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5.21 The density integral, which can be interpreted as a sort of mass in a two-dimensional setting, is plotted over time, together with the phase parameter. This plot shows how material flows in to compensate for shrinkage, but this inflow decelerates quickly due to solidification. . . 87 5.22 The average temperature and temperature range are shown in this plot. The

temperature comes very close to the lower limit 𝜃 = 300 after one half (𝑡 = 1) of the total simulation time. . . 87 5.23 The stress-field is shown at different points in time. Build up of deviatoric

stress can be observed especially near the inlet. . . 88 5.24 The stress is plotted over time. Some slight relaxation is visible near the

be-ginning. In the solidified material, the stress remains as residual stress. . . 88 5.25 The pressure and determinant of deformation, which is the volumetric part of

deformation, are shown in this plot. The pressure decreases since the solidified material is not free to contract as it cools down. . . 88

6.1 Sketch of injection molding machine and process. By

L. v. Lieshout, 2007, Retrieved December 7, 2019, from

https://commons.wikimedia.org/wiki/File:Injection_moulding_process.png . . 90 6.2 Overview of physical processes that occur in the material during injection

molding. . . 91 6.3 Overview of an approximation of the injection molding process that is suitable

for numerical simulation. Depicted are the physical phenomena that we allow during certain stages, in the top half of the figure, and the material behavior we account for, in the bottom half. . . 93 6.4 The full injection molding simulation takes a cavity shape and boundary

tem-perature as inputs calculates a molding shape, along with temtem-perature and residual stress fields. . . 93 6.5 We split the injection molding simulation into a solidification and a shrinkage

and warpage simulation. We make this split at the point in the process when the molding is ejected and keep the molding shape fixed until this point. . . . 94 6.6 The solidification simulation takes a cavity shape and boundary temperature

as input. It calculates the temperature and stress fields at ejection time. The shape is left unchanged. . . 94 6.7 The shrinkage and warpage simulation takes the ejection shape with

temper-ature and strain fields as input. From this it calculates the cooled molding shape with temperature and residual stress distribution. . . 95 6.8 If the solidification simulation is provided with an initial guess for the cavity

shape it will produce temperature and stress fields at ejection time. The in-verse shrinkage and warpage simulation, given a desired molding shape, can estimate an improved cavity shape. At the same time it will predict tempera-ture and stress fields at ejection time in this improved cavity shape, as well as a residual stress field in the molding shape. . . 96

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6.9 All different simulation types are shown along with their relations. Follow-ing the upper path, from the cavity shape on the left to the moldFollow-ing shape on the right, we obtain a forward injection molding simulation. If the forward shrinkage and warpage simulation is replaced by its inverse equivalent (bot-tom), we obtain a prediction for an improved cavity shape. If this improved shape is repeatedly fed back to the solidification simulation, an iteration loop is obtained. . . 97 6.10 Velocity and phase parameter fields are shown at 𝑡 = 0.1. There is still some

motion in the material, and it is not entirely solidified. The remaining flow — which can be compared to plastic deformation — is ignored in the elastic shrinkage and warpage simulation. The existence of a fluid phase in the mold-ing’s interior is normal in the injection molding process at ejection time. . . . 99 6.11 The initial and physical temperature fields are shown on the respective shapes

for the forward simulation. Due to the inhomogeneous initial temperature field, the body shrinks and the two ’arms’ tilt inwards. . . 99 6.12 The initial and physical stress fields are shown for the forward simulation.

There is a significant increase in stress due to shrinkage and warpage. . . 100 6.13 The inverse simulation yields different initial and physical shapes. The

ini-tial temperature field shown on the left-hand side is the estimate of how the temperature field could look in the adjusted shape. . . 100 6.14 The initial and physical stress fields are shown for the inverse simulation.

The residual stress field may help estimate if the body will undergo plastic deformation or even break. . . 100 6.15 The Hausdorff distance between the molding shape that is predicted by the

forward simulation and the desired shape is plotted after every iteration step. There is a huge drop in the distance after the first iteration step, and only minor changes afterwards. The distance in the second iteration (i.e., after just one step of the inverse elasticity solution) is smaller than the previous distance by a factor 18. The smallest distance value reached is approximately 0.01518, which is 0.38% of the reference length 4. The absolute value of the change (delta) between the distances of subsequent iteration steps is plotted on a logarithmic scale. The ongoing decrease in this delta suggests that the distance converges to some finite value. . . 101 6.16 This figure shows how the molding shape that results in the forward shrinkage

and warpage simulation (black) approaches the desired shape (green) with each iteration step. Significant zoom is required to show the shape differences in any but the first iteration steps. . . 102 6.17 This figure shows the changes in the cavity shape between iteration steps.

The cavity shapes that are predicted by the inverse shrinkage and warpage simulation in different iterations (black) are compared to the cavity shape in the last iteration (red), which yields the best molding shape in terms of the Hausdorff distance. Improvements in the cavity shape are still visible after the second iteration, but only with high zoom. . . 102

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6.18 On the left-hand side of this figure, the temperature field that is estimated in the inverse solution in Iteration 1 is compared to the field that actually results in the solidification simulation with the adjusted cavity shape in Iteration 2. An isoline is shown at 𝜙 = 320 to highlight the differences in the temperature field. Significant differences are visible. However, on the right-hand side, the temperature field that is estimated in the inverse solution in Iteration 2 is com-pared to the temperature field that results from the solidification simulation in the very last iteration. The differences between these fields are no longer visible. . . 103 6.19 The residual stress field that is estimated by the inverse shrinkage and

warpage simulation in the first iteration is compared to the stress fields cal-culated in Iterations 2 and 13. There are significant differences between the estimated and calculated stress fields. If the residual stress in the molding is important, the forward simulation needs to be run after any inverse simula-tion run to get more reliable results. However, there seem to be no significant changes in any of the further iteration steps. . . 103 6.20 Initial and physical temperature fields in forward mode. The deformation in

the body is similar to the deformation in the previous section, but with some additional effects near the inlet. . . 104 6.21 Initial and physical stress fields in forward mode. Stress peaks are seen near

the inlet. . . 104 6.22 Initial and physical shapes calculated in inverse mode. The initial temperature

field on the left-hand side is estimated by transforming the known tempera-ture field geometrically with the help of the smoothing equation. . . 105 6.23 Initial and physical stress fields in inverse mode. . . 105 6.24 The Hausdorff distance between the molding shape calculated in each

itera-tion’s forward simulation and the desired shape is shown in the plot. Addi-tionally, we show the absolute value of the change (delta) in the Hausdorff distance between the iteration steps on a logarithmic scale. As in the previ-ous example, the iteration appears to converge to a finite distance value. This time, however, more iteration steps are needed before the Hausdorff distance is close to its minimum value of 0.01854. . . 105 6.25 The molding shape (black) that is calculated in the forward simulation of each

iteration step is compared to the desired shape (green). The differences are large in Iteration 1, but zoom is required to properly show them in the follow-ing iteration steps. . . 106 6.26 The cavity shape (black) that is proposed in the inverse simulation of each

iteration step is compared to the cavity shape in the final iteration (red). The cavity shape in the first iteration is already close to the final result, but small improvements are made in the following iteration steps. . . 107

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List of Tables

4.1 The configuration of the simulation is summarized in this table. The deforma-tion is caused by gravitadeforma-tional forces. . . 39 4.2 Simulation parameters for the heat equation. The simulation is run for just a

single time step to achieve partial cooling to an inhomogeneous temperature field. . . 42 4.3 The simulation is started with zero initial temperature everywhere. The

boundaries are fixed at 𝜃 = −50 to cool down the material. . . 42 4.4 Constitutive laws and parameters in the thermoelasticity simulation. The

ex-ponential thermal expansion law is chosen to allow significant temperature changes. The thermal conductivity used in the Fourier heat flux law is of no physical consequence due to the uniform boundary conditions of the temper-ature equation. . . 43 4.5 The boundary conditions for the temperature are identical to those used in

the preparatory heat equation simulation. Zero boundary forces are applied via Neumann boundary conditions to allow the body to reach an equilibrium state. . . 43 5.1 General simulation settings and parameters. . . 79 5.2 Initial and boundary conditions of the simulation. The simulation starts with

a uniform temperature 𝜃 = 350 and a fluid at rest. It is cooled down at the boundaries to 𝜃 = 300. No-penetration boundary conditions are used for the momentum equation, to allow friction-free slipping of the material along the boundary. . . 79 5.3 Constitutive laws and parameters. The parameter values are chosen for

illus-trative purposes and are not meant to represent actual materials. . . 79 5.4 List of simulation parameters. . . 85 5.5 Initial and boundary conditions of the simulation. At time 𝑡 = 0, the fluid is at

rest at a uniform temperature. . . 85 5.6 Constitutive laws and parameters. . . 85 6.1 Constitutive laws and parameters used in the shrinkage simulation. The

con-stitutive laws and parameters have been chosen to be consistent with those used in the solidification simulation. The thermal conductivity has no effect on the solution in these conditions. . . 98

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6.2 Simulation settings and boundary conditions. The boundary are conditions for the temperature equation are identical to those used in the solidification simulation. Neumann boundary conditions for zero boundary forces are used for the momentum equation so the body can take on its equilibrium shape. . . 98

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Chapter 1 Introduction

When plastics parts are manufactured in an injection molding process, it is difficult to achieve the desired part shapes exactly. The part shapes — or molding shapes — are, in principle, prescribed through the mold cavity shape. However, the molding shape and cavity shape do not necessarily correspond exactly to one another (see Figure 1.1). One reason for this is the inhomogeneous cooling behavior in the polymer melt, which, in turn, leads to inhomogeneous solidification and shrinkage behavior. Apart from that, the molding is constrained by the cavity shape and cannot shrink freely. This causes further warpage in the molding.

Cavity Shape Molding Shape

Figure 1.1: The shape of a molding that is produced by injection molding differs from the cavity shape in the mold. These shape differences result from the cooling-induced material shrinkage and residual stresses caused by the solidification.

There are several approaches towards improving the molding shapes, i.e., bringing them closer to the desired result. Since the cavity shape has the greatest influence on the molding shape, certain shape changes that occur during the injection molding process can be compensated for in the cavity shape. Our goal is, therefore, to find the cavity shape that will produce the optimal molding shape, as depicted in Figure 1.2.

Of course, other aspects of the process influence the molding shape. Many methods have been devised to optimize, e.g., the process conditions [1–3] or the cooling channels [1, 4]. In order to achieve our specific goal, we are looking for a computational method that can determine a suitable cavity shape if a desired molding shape is provided. Such a method requires, above all, a mechanism that can predict the molding shape for any given cavity shape. This task is achieved by means of a numerical simulation of the injection molding process.

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Cavity Shape Molding Shape

Figure 1.2:The main objective of this document is to develop a method that can predict a cavity shape for a desired molding shape.

of the physical behavior of polymeric materials and a general outline of numerical simulation techniques in Chapter 2. The injection molding process involves the solidification of the liquid polymer melt and, therefore, both fluid and solid phases. While both of these phases can, in theory, be simulated at the same time — they actually have to be simulated simultaneously during the solidification process — it still makes sense to use separate formulations for the predominantly fluid and predominantly solid process stages. For this reason, we describe a stationary formulation of solid thermoelasticity in Chapter 3 and, additionally, a transient formulation of fluid and solid thermoviscoelasticity in Chapter 5. Both of the formulations are complete with a description of the simulation models and a discretization with the Finite Element Method. The formulations, including all differential equations and constitutive laws, are presented in a manner that lends itself to a swift implementation in a flexible simulation computer code. Finally, the specific application of injection molding and the mechanism of combining the two formulations are described in detail in Chapter 6.

With suitable simulation methods in place, the task that remains is the efficient determina-tion of a cavity shape that will lead to the desired simuladetermina-tion result. A common technique that comes to mind is mathematical shape optimization. In order to use such a method, an objective function must be defined that measures the quality of the simulation result. The task of the optimization method is to search for the minimum of this objective function. In our case, an objective function would basically have to translate geometric differences between calculated and desired molding shapes into a scalar value. The actual optimization could be accomplished with, e.g., quasi-Newton methods, such as BFGS (cf., e.g., [5]). Some examples of shape optimization techniques in polymer processing include [6–9], where these are applied to extrusion processes.

In this document, we discuss an alternative technique for the inverse design of the cavity shape, which does not rely on the minimization of an objective function. Specifically, we exploit the fact that the stationary formulation of solid thermoelasticity can be inverted easily (see Chapter 4). Based on this inverse formulation, we also propose an iterative technique for the cavity shape design (see Section 6.4).

The principal simulation methods, along with the inverse formulation and the iteration tech-nique for the determination of the cavity shape, are finally investigated in a few illustrative numerical examples (see Sections 4.4, 5.13, 6.5).

Throughout the document, we will use some specific mathematical notations. These are listed and explained in Appendix A.

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Chapter 2 Theoretical Background

Before we can start with the description of the simulation methods, some relevant theoretical aspects concerning physics, modelling, and numerics have to be explained.

2.1 Polymer Physics

The purpose of this section is to explain some general physical properties of polymers that will be relevant for motivating and understanding the methods that are described in this doc-ument. We will limit ourselves to purely isotropic materials and will therefore not discuss any composite materials such as fiber-reinforced polymers.

Polymers, generally speaking, are made of molecules that are chains of repeating units (cf., e.g., [10, p. 1]). We will exclusively discuss the behavior of carbon-based polymers. Our focus in this section is on the different states of polymers and the solidification processes. For a more detailed overview of polymer physics, we refer to Gedde [10], Strobl [11] and Ward and Sweeney [12].

2.1.1 Polymer States and Behavior

The behavior of polymers is extremely variable, depending completely on certain conditions or the state of the material. If a polymer’s behavior is observed at different temperatures but otherwise similar conditions, one can see multiple types of behavior. At very low tem-peratures, the polymer behaves like a glassy or brittle solid. This means it has a very high elastic modulus and can break at very small strains. At increased temperatures, the polymer behaves like a rubber. In this state, the elastic modulus may be decreased by a factor of about one thousand and the material can withstand large strains without breaking. At even higher temperatures, the polymer behaves like a fluid with a high viscosity (cf. [12, p. 19]).

If, on the other hand, a polymer’s behavior is observed at a fixed temperature, but at different measurement frequencies or time scales, all of these behavior types may still occur. In fact, raising the temperature changes the observation in a similar way as decreasing the frequency of measurements. The consequence of these findings is that a polymer is difficult to classify

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as either a solid or a fluid, even if only a fixed temperature is considered. Instead it needs to be seen as something in between. This is called viscoelastic material behavior.

The similarity of temperature changes to frequency changes is called time-temperature equiv-alence (cf., e.g., [12, p. 135]). It motivates some empirical relationships between material pa-rameters and the temperature, such as the Williams-Landel-Ferry (WLF) equation (see [13, p. 28]).

2.1.2 Stress Relaxation

The most important aspect of viscoelastic material behavior is the relaxation of stresses. If a viscoelastic material is put under shear strain, the resulting stress decreases with time. The speed at which this happens varies. The characteristic time of this process is called relaxation time. It is possible to associate a continuous spectrum of relaxation times to a material (cf. [12, p. 101]).

The concept of time-temperature superposition also applies for this relaxation time. Simple relationships between relaxation time and temperature are often proposed on this basis (cf., e.g., [14]).

We will discuss modeling approaches for viscoelasticity and stress relaxation in Section 2.2.4.

2.1.3 Types of Polymers

At high temperatures, the long molecule chains in the polymer are entangled in a chaotic fashion. In this state, the polymer is called amorphous. In some polymers, a part of the chains align when the material is cooled down. This is called crystallization. Since an amorphous phase always remains in the material, the polymers capable of crystallization are called semi-crystalline (cf., e.g., [11, p. 166], [15, p. 19]).1

While both amorphous and semi-crystalline polymers behave similarly in a molten state, where they are both amorphous, their solidification behavior is completely different.

2.1.4 Glass Transition

When an amorphous polymer is cooled down from a hot, molten state, both the material behavior and the specific volume undergo changes. The same applies for the amorphous phase in semi-crystalline polymers. We have already stated that the material behavior of polymers at low temperatures appears more glass-like, and the elastic modulus is very high, whereas at high temperatures, the material behavior is closer to that of a fluid, and the elastic modulus is much lower. These aspects, however, depend on the observation frequency, as stated before. Another property that shows a dependence on the temperature is the specific volume. This decreases with decreasing temperature, such that the material shrinks during cooling. In certain ranges of temperature or frequencies, these changes are gradual. However, there is one point of rapid change. This is called the glass transition. At the glass transition, the elastic 1_{Many polymers can be categorized as fully amorphous or semi-crystalline, but there are also others. See,}

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modulus can suddenly change by a factor of 1000 (cf. [10, p. 14]). While there is no sudden change in the specific volume, the rate at which this changes with the temperature experiences a jump (cf. [10, p. 83]). The specific volume is less sensitive to temperature changes below the glass transition temperature.

When speaking of this glass transition temperature, it should be noted that this is not a fixed material property. To the contrary, this temperature depends completely on the cooling rate and other conditions. It has been found that an increase in the cooling rate also leads to an increase in the glass transition temperature (cf. [10, p. 83]). In other words, faster cooling leads to a much earlier glass transition. This behavior is still not well understood and is subject of ongoing research (cf. [16]).

The glass transition is sometimes compared to a second-order thermodynamic phase transi-tion. This would mean that there is no jump in the specific volume as the temperature de-creases. This may be accurate for constant cooling rates, but if, for instance, the temperature is suddenly held constant early after the glass transition, the specific volume keeps decreas-ing with time. This effect is called physical agedecreas-ing and is equivalent to a jump in the specific volume with respect to temperature changes (cf. [10, p. 230]).

When it comes to fully understanding and modeling the glass transition, there are still a lot of unanswered question (cf., e.g., [17]). Simple models are available, however, to estimate the specific volume as a function of temperature across the glass transition (cf., e.g., [18]).

2.1.5 Crystallization

Besides the glass transition, there is another solidification process that can occur in polymers. This is the crystallization. As mentioned above, this only occurs in certain polymers, which we call semi-crystalline polymers. Whether crystallization occurs, and how fast it happens, depends, among other things, on the molecular mass of the polymer. While materials of lower molecular mass crystallize immediately and rapidly when they are cooled below their melting point, polymers, which generally have a high molecular mass, do so much less readily, if at all.

Crystallization in polymers is a process where the polymer chains start to fold and align. This process is split into two principal components, which are nucleation — i.e., the formation of small crystal nuclei — and subsequent crystal growth. Nucleation can happen both instanta-neously when, e.g., a certain temperature threshold is reached, or sporadically over time. In any case, nucleation only starts below the polymer’s melting temperature and, in many cases, only at significant undercooling (i.e., far below the melting temperature). Once a nucleus has formed, it starts to grow. This means that more polymer chains start to fold and align with the crystallite. The rate of this crystal growth also depends on the temperature. Most impor-tantly, both nucleation and crystal growth slow down as the glass transition temperature is approached, and subside altogether when this is reached. Overall, polymer crystallization is a slow process that happens in a large temperature range between the melting temperature and the glass transition temperature. We refer to, e.g., Gedde [10, p. 169], Mandelkern [19, p. 65], or Strobl [11, p. 181], for more detailed information on this topic.

For particular interest in polymer processing and, particularly, injection molding, is the effect that crystallization has on the material properties. Under quiescent conditions, i.e., slow or

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negligible flow, the crystallites that grow from the nuclei do so in an isotropic, or radial, manner. The resulting structures are called spherulites. If, on the other hand, nuclei are subject to shear flow, they can grow into threads. The crystallite structures that grow from these nuclei take on a different shape that is sometimes referred to as shish-kebab. In contrast to spherulites, these structures do not have a homogeneous distribution of chain orientations: they are anisotropic. The occurence of this so-called flow-induced crystallization is of great significance in injection molding, since the crystal anisotropy also leads to anisotropy in the shrinkage and warpage behavior of the material. More information on the relevance of these crystallization processes for injection molding can be found in, e.g., Zheng [13, p. 47] or Kennedy and Zheng [15, p. 141].

The task of modeling polymer crystallization in a numerical simulation can be approached from several points or, more exactly, on multiple scales. A popular approach for modeling crystallization of various materials on a microscopic scale is the phase-field method (see, e.g., [20]). This method has been applied specifically to model spherulite growth by Gránásy [21]. There are also macroscopic equations that are used to approximate the nucleation and crystal growth behavior. A well-known theory for the radial growth of spherulites is the one proposed by Lauritzen and Hoffman [22].

2.2 Rheology

In order to discuss the rheology of polymers, which behave viscoelastically, it makes sense to first mention simplified models for the material behavior of pure fluids and solids. We focus here on the material behavior with respect to mechanical forces. In this section, we will compare the different models using one-dimensional differential equations, for simplicity. Corresponding three-dimensional equations will be provided in Chapters 3 and 5.

2.2.1 Hookean Solids

A very simple example of a solid material law is Hooke’s law. This was originally formulated for an elastic spring, such as that depicted in Figure 2.1. It proposes that the force due to deformation is proportional to the change in length of the spring (see, e.g., [23, p. 5], [24, p. 63]). 𝜇 −0.20 0.2 0.4 0.6 0.81 1.2 1.4 −2 0 2 4 6 8 10 Strain 𝛾 Time 𝑡 Hooke

Figure 2.1: An elastic spring with stress 𝜏 and strain (i.e., rel-ative length change) 𝛾.

Figure 2.2:The strain 𝛾 is shown if a constant stress ̄𝜏 = 1 is sud-denly applied at 𝑡 = 0 to a Hookean solid with modulus 𝜇 = 1.

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In this section we formulate material laws that relate a shear stress 𝜏 and shear strain 𝛾. These can be roughly compared to, for instance, the off-diagonal components in symmetrized stress and strain tensors in two dimensions. For the elastic spring we can formulate the stress-strain relationship

𝜏 = 𝜇𝛾 , (2.1)

with an elasticity modulus 𝜇.

In order to analyze this material law a bit further, we observe its reaction to a suddenly applied constant stress ̄𝜏, i.e.,

𝜏(𝑡) = 𝐻(𝑡) ̄𝜏 , (2.2)

with the Heaviside step function 𝐻(𝑡).2 _{The resulting strain 𝛾(𝑡) = 𝐻(𝑡) ̄𝜏/𝜇 is plotted in} Fig-ure 2.2.

If the shear deformation of a material can be described with this stress law, the only stress-free state is the one with zero deformation. All the mechanical work that is put into the body’s shear deformation is stored as free energy and can be released completely as the body returns to its original state.

2.2.2 Newtonian Fluids

A Newtonian fluid is assumed to behave completely different from a Hookean solid (cf., e.g., [23, p. 65] or [24, p. 64]). Most importantly, none of the shear deformation work is stored as free energy. Instead, this energy dissipates and produces entropy. This corresponds to a stress-strain relationship of

𝜏 = 𝜂 ̇𝛾 , (2.3)

with viscosity 𝜂. ̇𝛾 is used for the time rate of the strain, or simply strain rate.

Similar to the Hookean solid, we compare the Newtonian fluid to a simple mechanical model. In this case, this is a dashpot, as depicted in Figure 2.3.

𝜂 −20 2 4 6 8 10 12 −2 0 2 4 6 8 10 Strain 𝛾 Time 𝑡 Newton

Figure 2.3: A viscous dashpot

with stress 𝜏 and strain 𝛾. Figure 2.4:denly applied at 𝑡 = 0 to a Newtonian fluid with viscosity 𝜂 = 1.The strain 𝛾 is shown if a constant stress ̄𝜏 = 1 is sud-When a Newtonian fluid is suddenly exposed to a constant stress, same as the Hookean solid in the previous section, the resulting strain function is

𝛾(𝑡) = 𝐻(𝑡) ̄𝜏𝜂𝑡 . (2.4)

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A plot of this function is shown in Figure 2.4.

2.2.3 Generalized Newtonian Fluids

In polymeric fluids, among others, a behavior can be observed where the viscosity decreases at high shear rates. This is called shear thinning (see, e.g., [24, p. 29]). This type of behav-ior cannot be represented with the Newtonian fluid law. As a simple generalization to the Newtonian fluid, the viscosity is sometimes formulated as a function of the shear rate. These laws are called generalized Newtonian laws. One such law that is used to approximate the shear-thinning behavior of polymers is the Carreau model (see [13, p. 18]).

In analogy to the one-dimensional stress-strain relationships given so far, we could formulate a law similar to the generalized Newtonian fluid as

𝜏 = 𝜂( ̇𝛾) ̇𝛾 , (2.5)

with 𝜂( ̇𝛾) a monotonically decreasing function of the shear rate, in case of shear thinning.

2.2.4 Viscoelastic Fluids

While the generalized Newtonian fluid may be a suitable approximation for polymeric fluids at very high temperatures or very low observation frequencies, they do not consider the elastic behavior of polymeric fluids that is relevant in other conditions. As the term ’viscoelastic’ already suggests, we are looking for a material law that mixes aspects of both viscous and elastic, i.e., fluid and solid, material behavior.

Maxwell The Maxwell model is a simple model that fulfills these requirements. In analogy with the solid and fluid laws, we can compare it to a model of simple mechanical components. In this case, an elastic spring, as in the Hookean solid law, and a viscous dashpot, as in the Newtonian fluid law, are combined in series (cf., e.g., [12, p. 99]). This is depicted in Figure 2.5.

𝜇 𝜂 −0.20 0.2 0.4 0.6 0.81 1.2 1.4 −2 0 2 4 6 8 10 Strain 𝛾 Time 𝑡 Hooke 𝜇 Maxwell −2 0 2 4 6 8 10 12 −200 0 200 400 600 800 1000 Strain 𝛾 Time 𝑡 Newton 𝜂 Maxwell

Figure 2.5: The Maxwell fluid model is comparable to an elastic spring and vis-cous dashpot combined in series.

Figure 2.6: The strain 𝛾 caused in a Maxwell fluid by a stress ̄𝜏 = 1 that is suddenly applied at 𝑡 = 0 is shown. The viscosity is chosen as 𝜂 = 100 and the modulus as 𝜇 = 1. The plot is shown for a shorter (left) and a longer (right) observation time. Hookean solid and Newtonian fluid behavior are shown for comparison. In the beginning the curve is similar to that of a Hookean solid, but tends to that of a Newtonian fluid in the long term.

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The stress-strain relationship of this law can be represented through the first-order differential equation

̇𝜏 + 𝜇𝜂𝜏 = 𝜇 ̇𝛾 . (2.6)

This is an equation for the stress that only requires the strain rate ̇𝛾 and no absolute strain measure. This is consistent with material laws in the Eulerian frame that are formulated by means of the velocity field (see Chapter 5). A three-dimensional Maxwell model actually requires a more complicated time derivative to guarantee the stress tensor objectivity: the so-called upper convected derivative or Oldroyd derivative (see, e.g., [24, p. 130]). This will also be used in our Eulerian formulation (see Section 5.4.2).

If a constant stress ̄𝜏 is applied to a Maxwell fluid, it reacts with the shear strain and shear rate

𝛾(𝑡) = 𝐻(𝑡) ̄𝜏 (1𝜇 +𝜂𝑡) ,1 ̇𝛾(𝑡) = 𝐻(𝑡)1𝜂 ̄𝜏 . (2.7) These functions can tell us that the material initially reacts to the stress similar to a Hookean solid with shear modulus 𝜇, since

𝛾(0) = 1𝜇 ̄𝜏 . (2.8)

This behavior is also visible in the plot in Figure 2.6 (left), where the material is compared to a Hookean solid. In the long term, the material behaves like a Newtonian fluid with viscosity 𝜂, since

lim

𝑡→∞ ̇𝛾(𝑡) = 1𝜂 ̄𝜏 . (2.9)

The corresponding strain plot, along with a comparable Newtonian fluid, is shown in Fig-ure 2.6 (right).

In a different scenario, where a constant strain is applied, the stress relaxes towards zero. This is evident from

̇𝛾 = 0 ⇒ _{̇𝜏 = −𝜇𝜂𝜏 ⇔ 𝜏 = 𝜏}0exp (−𝜇_{𝜂𝑡) ,} (2.10)

with 𝜏 = 𝜏0at 𝑡 = 0. Here, we call 𝜆 = 𝜂/𝜇 the relaxation time.

At higher frequencies, the shear rate ̇𝛾 dominates. In this regime the material behavior is similar to that of an elastic solid, and most of the shear deformation work is stored as free energy and released again before it dissipates. At low frequencies, however, the shear rate ̇𝛾 becomes negligible, and the material approaches the behavior of a viscous fluid, i.e., the energy due to shear deformation work dissipates almost completely. If we compare this to the polymer behavior types at different observation frequencies that we discussed in Section 2.1.1, we can conclude that the Maxwell model can represent the behavior between the rubber elastic state and the viscous state. This means, on the other hand, that this model does not represent the glassy behavior at very high frequencies or very low temperatures. These properties underline the suitability of this model for viscoelastic fluids, well above the glass transition temperature, rather than solids.

As mentioned in Section 2.1.2, the stress relaxation process cannot be described by means of a single relaxation time. Instead, there is a whole spectrum of relaxation times. A suitable model

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to approximate this behavior would have to include multiple spring-dashpot combinations (see Figure 2.7).

Figure 2.7:Spring-dashpot combinations as in the Maxwell model are com-bined in parallel to model multiple relaxation times.

𝜇1 𝜂1 𝜇2 𝜂2 𝜇3 𝜂3 𝜇𝑛 𝜂𝑛

The stress can now be calculated as the sum of the stresses in the individual spring-dashpot combinations: 𝜏 = ∑𝑛 𝑖=1 𝜏 ′ 𝑖 , (2.11) ̇𝜏′ 𝑖 + 𝜇_𝜂𝑖 𝑖𝜏 ′ 𝑖 = 𝜇𝑖 ̇𝛾 . (2.12)

The relaxation times of the individual modes 𝜆𝑖 can be adjusted through the moduli 𝜇𝑖 and viscosities 𝜂𝑖 such that 𝜆𝑖 = 𝜂𝑖/𝜇𝑖. The elasticity moduli 𝜇𝑖 can be interpreted as weights for the individual modes with respect to the system’s elastic modulus. Analogously, the viscosities 𝜂𝑖 are weights with respect to the system’s viscosity.

Oldroyd-B In practice, the Maxwell model is often extended by adding another viscosity term. The result is the Oldroyd-B model (cf. [25]). In the analogous spring-dashpot model, this change is achieved by adding a parallel dashpot, as in Figure 2.8.

Figure 2.8: A mechanical model that is similar to the Oldroyd-B material is ob-tained by adding a dashpot in parallel to the spring-dashpot combination from the

Maxwell model. _𝜇_′ _𝜂_′

𝜂

This model leads to the same differential equation as the Maxwell model, but this time only for a part of the stress, the extra stress. The complete stress is the sum of extra stress and the additional viscous stress:

𝜏 = 𝜂 ̇𝛾 + 𝜏′ _, _(2.13)

̇𝜏′_{+ 𝜇}′

𝜂′𝜏′ = 𝜇′ ̇𝛾 . (2.14)

In this form, the model can be compared to the three-dimensional model in Section 5.9.6. It can also be formulated without the extra stress using the second-order differential equation

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In this model, we call the additional viscosity 𝜂 the solvent viscosity and 𝜂′ _{the polymeric} viscosity. −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 −0.02 0 0.020.040.060.08 0.1 Strain 𝛾 Time 𝑡 Newton 𝜂 Oldroyd-B −0.20 0.2 0.4 0.6 0.81 1.2 1.4 −2 0 2 4 6 8 10 Strain 𝛾 Time 𝑡 Hooke 𝜇′ Maxwell Oldroyd-B −20 2 4 6 8 10 12 −200 0 200 400 600 800 1000 Strain 𝛾 Time 𝑡 Newton 𝜂 + 𝜂′ Maxwell Oldroyd-B

Figure 2.9: The strain 𝛾 is plotted over time when a stress ̄𝜏 = 1 is suddenly applied to an Oldroyd-B fluid at 𝑡 = 0. The viscosities are set to 𝜂 = 1 and 𝜂′ _{= 99, and the modulus to 𝜇 = 1. Newtonian, Hookean, and}

Maxwellian materials are shown for comparison. At first, the material behaves viscously (left), later it shows similarities to elastic behavior (middle), before approaching fluid behavior again in the long term (right). We apply a constant stress ̄𝜏 to the Oldroyd-B fluid to compare its behavior to the previous materials. This yields the strain function3

𝛾(𝑡) = 𝐻(𝑡) ̄𝜏_𝜂_′ _{+ 𝜂 (}_𝜇_′_(𝜂𝜂_′′𝜂_{+ 𝜂) (1 − exp (−}′ 𝜇′(𝜂_𝜂′_′_{𝜂 𝑡)) + 𝑡) .}+ 𝜂) (2.16) In contrast to the Maxwell fluid, the Oldroyd-B fluid initially reacts like a viscous fluid. The initial viscosity is 𝜂. We obtain

𝛾(0) = 0 , _𝑡→0lim₊ _{̇𝛾(𝑡) = 1𝜂 ̄𝜏 .} (2.17)

This behavior is also visible in Figure 2.9 (left). In an intermediate time range, the strain curve of the Oldroyd-B fluid looks similar to that of a Hookean solid with shear modulus 𝜇 (see Figure 2.9 (middle)). This suggests that this material law is capable of representing the rubber plateau that was described in Section 2.1.1. In the long term, the material behavior tends to that of a Newtonian fluid again, this time with the higher viscosity 𝜂′_{+ 𝜂:}

lim

𝑡→∞ ̇𝛾(𝑡) = 1

𝜂′_{+ 𝜂 ̄𝜏 .} (2.18)

See Figure 2.9 (right) for the long-term strain curve of the Oldroyd-B fluid compared to a Newtonian fluid.

Overall, we can conclude that the material behavior of Oldroyd-B and Maxwell fluids is similar at low observation frequencies. In the high-frequency range, the Oldroyd-B model is capa-ble of showing some phenomena that are missing in the Maxwell model, such as the rubber plateau. A direct comparison of the strain curves is shown in Figure 2.10.

The Maxwell and Oldroyd-B models are very simple models that have been presented here to introduce the topic of fluid viscoelasticity. Some models of more practical relevance are, e.g.,

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Figure 2.10: Strain curves for Maxwell (𝜂 = 100, 𝜇 = 1) and Oldroyd-B (𝜂 = 1, 𝜂′ _{= 99, 𝜇}′ _{= 1) fluids are shown when a}

stress ̄𝜏 = 1 is suddenly applied at 𝑡 = 0. Due to the logarithmic plot, all the differ-ent behavioral regimes are visible, includ-ing the difference in behavior at high fre-quencies. There is a clearly visible plateau — the rubber plateau — in the Oldroyd-B

curve. 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 0.01 0.1 1 10 100 1000 10000 Strain 𝛾 Time 𝑡 Hooke 𝜇′ Newton 𝜂 Newton 𝜂 + 𝜂′ Maxwell Oldroyd-B

the Giesekus model (see Section 5.9.6 and [26]) or the Phan-Thien-Tanner model (cf. [27]). For a more complete overview of polymer rheology, we refer to Phan-Thien [24] or Macosko [23]. Viscoelastic flow is often characterized by the so-called Weissenberg number. Given a charac-teristic shear rate ̇𝛾, this is defined as (see, e.g., [24, p. 66])

Wi = 𝜆 ̇𝛾 . (2.19)

The Weissenberg number is of great significance in numerical implementations of this model. It has been found that high Weissenberg numbers cause issues with numerical stability (cf. [28]). High Weissenberg numbers are obtained especially if long relaxation times and a large shear rate coincide. The former occurs when the material behavior is close to rubber elasticity. Some recent advances and studies with respect to the High Weissenberg Number Problem can be found in [29].

2.2.5 Viscoelastic Solids

The viscoelastic fluid models presented in the previous section have the property that stresses relax towards zero in the long term. As mentioned before, the behavior of polymers at low temperatures or high frequencies varies between glassy and rubbery elastic. Complete stress relaxation does not play a role in this case and can be neglected in the model.

Kelvin-Voigt We start the description of solid viscoelasticity with the Kelvin-Voigt model (cf. [12, p. 98]). This model is a simple combination of elastic and viscous material behavior that leads to a model without stress relaxation. In analogy to the previous model descriptions, Figure 2.11 contains a dashpot model that represents a Kelvin-Voigt solid. The spring-dashpot model indicates that the Kelvin-Voigt solid is identical to an Oldroyd-B fluid with vanishing polymeric viscosity. More importantly, it is the limit of the Oldroyd-B model as the relaxation time tends to infinity.

The stress in the Kelvin-Voigt solid can be calculated using a single first-order differential equation,

𝜏 = 𝜏′_{+ 𝜂 ̇𝛾 ,} _(2.20)

(31)

𝜂 𝜇 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 −0.02 0 0.020.040.060.08 0.1 Strain 𝛾 Time 𝑡 Newton 𝜂 Kelvin-Voigt −0.20 0.2 0.4 0.6 0.81 1.2 1.4 −2 0 2 4 6 8 10 Strain 𝛾 Time 𝑡 Hooke 𝜇 Oldroyd-B Kelvin-Voigt

Figure 2.11: The Kelvin-Voigt solid can be compared to a mechnical model with an elastic spring and a vis-cous dashpot in parallel.

Figure 2.12: The strain 𝛾 is shown if a stress ̄𝜏 = 1 is suddenly applied to a Kelvin-Voigt solid at 𝑡 = 0. The viscosity is chosen as 𝜂 = 1 and the modulus as 𝜇 = 1. Both short-term behavior (left) and longer-term behavior (right) are shown, in comparison to Newtonian, Hookean, and Oldroyd-B materials. The material at first reacts like a viscous fluid, but then tends to elastic material behavior in the long term.

if only the shear rate ̇𝛾 is available. Analagous to the Oldroyd-B model, there is the extra stress 𝜏′_{. In this case, the differential equation is missing the relaxation term. The model can} also be represented by a single differential equation, if the shear strain 𝛾 is available:

𝜏 = 𝜇𝛾 + 𝜂 ̇𝛾 . (2.22)

In the constant stress experiment that was introduced in the previous sections, the Kelvin-Voigt solid reacts with the strain4

𝛾(𝑡) = 𝐻(𝑡) ̄𝜏𝜇 (1 − exp(−𝜇𝜂𝑡)) . (2.23)

Initially, the Kelvin-Voigt solid reacts like a Newtonian fluid with viscosity 𝜂:

𝛾(0) = 0 , _𝑡→0lim₊ _{̇𝛾(𝑡) = 1𝜂 ̄𝜏 .} (2.24)

This behavior is plotted in Figure 2.12 (left). In the long term, the behavior tends to that of a Hookean solid with shear modulus 𝜇. We obtain the shear strain

lim

𝑡→∞𝛾(𝑡) = 1𝜇 ̄𝜏 . (2.25)

In Figure 2.12 (right), the long-term strain curve is compared to that of a Hookean solid. An overview of the behavior of the Kelvin-Voigt model in comparison with the Oldroyd-B model is provided in Figure 2.13. This shows that the high-frequency behavior types are similar. Standard Linear Solid For completeness sake, we also mention the SLS, or Standard Linear Solid, model (see, e.g., [12, p. 100], for more information). This model results from adding an elastic spring in parallel to a Maxwell model, as depicted in Figure 2.14. We obtain a solid that behaves similar to the Kelvin-Voigt solid at lower frequencies, but has an initial elastic

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