Mechanical characterization and modeling of curing thermosets

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Mechanical Characterization and Modeling

of Curing Thermosets

Proefschrift

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 9 januari 2006 om 13.00 uur

door

Cornelis VAN ’T HOF

werktuigkundig ingenieur

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Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. L.J. Ernst

Toegevoegd promotor: Dr.ir. K.M.B. Jansen Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof.dr.ir. L.J. Ernst, Technische Universiteit Delft, promotor

Dr.ir. K.M.B. Jansen, Technische Universiteit Delft, toegevoegd promotor Prof.dr.ir. G.Q. Zhang, Technische Universiteit Delft

Prof.dr.ir. R. Marissen, Technische Universiteit Delft Prof.ir. A. Beukers, Technische Universiteit Delft Prof.dr.ir. R. Akkerman, Universiteit Twente Ir. R. Bressers, Philips Nijmegen, adviseur

ISBN-10: 90-9020262-5 ISBN-13: 978-90-9020262-4

Copyright c° 2005 by Cornelis van ’t Hof

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy-ing, recording or otherwise, without the prior written permission of the publisher. Cover ideas realized by A. van ’t Hof

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List of Symbols

Subscript indices are often used to discriminate discrete values of a variable.

α Degree of cure, name of primary/glass-rubber transition −

β Name of sub-Tg relaxation −

γ Name of sub-Tg relaxation −

δ Phase angle between stress and strain sines in DMA ◦

δ0 _{Dirac function} ₋

∆ (t) Step function −

∆ Used to represent an increment, e.g. ∆t −

², ²ij Strain, strain tensor component −

²0 Strain amplitude −

²d _{Deviatoric strain tensor} ₋

²v _{Volumetric strain tensor} ₋

ζ Tolerance −

η Viscosity Pa s

θ Elapsed time after load application, t − ξ s

λ Lam´e constant Pa

λ0 _{Material constant in DiBenedetto relation} ₋

µ Lam´e constant Pa

ν Poisson’s ratio −

ξ, ξi Load application time s

σ, σij Stress, stress tensor component Pa

σ0 Stress amplitude Pa

σd _{Deviatoric stress tensor} _Pa

σh _{Hydrostatic stress tensor} _Pa

σt Transient stress component in DMA response Pa

σdecay

n Decaying part of the nth stress contribution Pa

σelast

n Instantaneous, elastic part of the nth stress contribution Pa

τn, τm Relaxation time s

τc Characteristic relaxation time s

τp Peak relaxation time s

τAR

n etc. Specific relaxation time s

ω Angular excitation frequency rad s−1

ωexp Constant intermittent cure excitation frequency rad s−1

ωact

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List of Symbols - Continued

a (T, T0) Shift function or factor for relaxation times −

a (T, α) Temperature and conversion dependent shift function

for relaxation times −

Ai Crosssectional area m

d, d0 Displacement, displacement amplitude m

E (θ) 1D relaxation modulus (Maxwell element) Pa

Ee Stiffness constant Maxwell element Pa

En Relaxation strength Pa

E∞, Erubber Rubbery part of 1D relaxation (Young’s) modulus Pa

Ea Activation energy J mol−1

F, F0 Force, force amplitude N

f0 _{Monomer functionality} ₋

gn Shear relaxation strength Pa

G Shear (relaxation) modulus Pa

Gn Shear relaxation strength Pa

Gg, Gglass Glassy shear modulus Pa

G∞, Grubber Rubbery shear modulus Pa

Gd Decaying part of the relaxation modulus Pa

G (θ) Shear relaxation modulus Pa

G0 _{Storage shear modulus} _Pa

G00 _{Loss shear modulus} _Pa

Gc_{, G}f _{Fully cure dependent shear relaxation function} _Pa

Gp _{Partly cure dependent shear relaxation function} _Pa

∆G Total relaxation strength, i.e. Gg− G∞ Pa

G0_|

n Storage model relaxation (Prony term) Pa

G00_|

n Loss model relaxation (Prony term) Pa

Gabs Complex shear modulus Pa

G0

n Alternative shear relaxation strength Pa

˜

G (θ, αi) Normalized shear relaxation modulus −

H, Hn Arrhenius activation energy J mol−1

Hr Residual exothermic heat of curing reaction J

Ht Total exothermic heat of curing reaction J

J (θ) Creep compliance Pa−1

k Boltzmann’s constant J K−1

kef f, kc, kd, k0 Rate constants s−1

K (θ) Bulk relaxation modulus Pa

K0 _{Tool stiffness} _{N m}−1

Kn Bulk relaxation strength Pa

Kg Glassy bulk modulus Pa

K∞, Krubber Rubbery bulk modulus Pa

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List of Symbols - Continued

R Gas constant J mol−1K−1

Rn(t, ξ0) Stress decay function −

Rc

n(t, ξ0) Ongoing cure stress decay function −

s Laplace variable −

Sn(ξ0) Stress scale factor Pa

Sc

n(t, ξ0) Ongoing cure stress scale factor Pa

t Current time s

t0 _{Curing time} _s

t000 _{Elapsed time after stopping harmonic excitation in DMA} _s

t0 Initial stress-free time s

tgel Curing time at which gelation occurs s

T Temperature K,◦_C

Tg Glass transition temperature K

Tcure Curing temperature K

Tg,0 Initial glass transition temperature (before cure) K

Tg∞ Ultimate glass transition temperature (at full cure) K

Tg,DMA Tg as determined by DMA K

Tg,DSC Tg as determined by DSC K

Tm Melting temperature of crystalline materials K

T∞ WLF parameter K

Ta Vogel-Fulcher parameter K

Tv Vogel-Fulcher parameter K

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List of Abbreviations

CPE Cure progress experiment

DMA Dynamic mechanical analysis

DS Dielectric spectroscopy

DSC Differential scanning calorimetry

FCD Fully cure dependent

FE Finite element

FRP Fiber reinforced polymer

FTIR Fourier transform infrared

H2O Water

HCl Hydrochloric acid

PCD Partly cure dependent

PVT Pressure - volume - temperature

SBT Sandwich beam tool

SST Simple shear tool

TCS Time-cure superposition

TTS Time-temperature superposition

TVCE Thermo-viscoelastic characterization experiment

UV Ultra-violet

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Preface

This thesis largely deals with stress relaxation in curing polymeric resins. In the following historical anecdote I will share my view on relaxation in polymers. Imagine a polymer consists of four brothers, placed on the back seat of a midsize car, some 20 years ago. No children’s safety seats were available at that time, just all four of them on one back seat. Of course, that isn’t a comfortable seat, initially. Striking arms and legs and a continuous struggle about the positions taken in by the brothers can be observed. You guessed already, this is a stressed situation. But now the car heads for Austria or another country full of mountains. After a while, some rearrangement can be observed on the back seat. Some sit back, others lean forward and on average the struggle becomes less. You guessed again, the stressed situation starts to relax! In this case both physically and mentally, and both on the back seat and the driver’s seat.

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Chapter 1 Introduction

In thermosetting resins the low molecular weight molecules form a 3D polymeric network while being (thermally) polymerized or cured. Opposite to thermoplastics, thermosetting polymers therefore cannot be molded after production and are thus polymerized in or around a structure, together forming the final product.

The fact that thermosets are applied as viscous resins offers several advantages relative to thermoplastics. The possibility of molding and subsequently curing allows thermosets to be used as complex-shaped, structural components. In case of low initial viscosity, even molding in or around fragile constructions is possible, as is important in for ex-ample the electronics industry. Low initial viscosity also allows impregnation of woven fabrics; polymer resins are widely used as matrices in composites, i.e. fiber-reinforced polymers (FRPs). Applications are found in the automotive, aerospace and marine in-dustries. Other applications of thermosets are as adhesives, paints, coatings and dental restorations.

The formaldehyde-based resins, developed by Baekeland around 1905 and better known as Bakelite, were the first thermosetting polymers successfully commercialized [50, 61]. Nowadays, global thermoset resin consumption stands at roughly 22 × 109_[kg]

a year. The four largest contributions are formed by polyurethanes (±34%), urea-formaldehyde (±32%), phenol-urea-formaldehyde (±15%) and unsaturated polyesters (±9%) [51]. The contribution of epoxies, as is dealt with in this thesis, is approximately 6% [5]. Epoxies have quite some advantages relative to other thermosets; they are very versa-tile, with excellent adhesive characteristics and excellent thermal, mechanical, chemical and electrical properties [10, 38, 45]. As a result, epoxies are often used for high perfor-mance applications.

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1. Residual stresses may exceed the fracture strength of the thermoset itself. In FRPs some fiber arrangements are seen to form constraints such that local resin fracture appears, thereby reducing maximum performance of the FRP already during the manufacturing stage. This problem was addressed by Kiasat [32]. Similar problems are experienced in curing concrete, see for example [41].

2. Also the adhesion strength of the interface with the surroundings may be exceeded. This occurs in dental restorations, where marginal leakage may be the result, re-sulting in turn in discoloration, tooth sensitivity and secondary caries [59].

3. Residual stresses may also cause failure of the surrounding structure. In electronic packages, for example, thermosets are often used to protect delicate parts from external influences or as structural components with a complex shape. It is here that residual stresses or strains cause failure of the package, for example by the loss of electrical contact. Sometimes breakdown appears already during curing, i.e. in the manufacturing stage, in other cases lifetime is reduced. In dental restorations residual stress may cause enamel fracture or even tooth cracking [40].

4. Finally, the residual strains itself may be a problem. Some FRPs will show warpage after curing. According to [44] warpage is a major obstacle for efficient and cost-effective manufacture of composite parts. This kind of dimensional inaccuracies may be critical in subsequent production processes, think for example of the assembly of body panels in cars. Curing induced shrinkage finally also diminishes the aesthetical appearance of this kind of panels, by sink marks in between the FRP fibre bundles. For product designers, it is important to have insight in the development of residual stresses and strains during curing. Only then one is able to adapt either the curing process, the material specifications or the product geometry such that shrinkage does not longer cause one of the above mentioned problems. Modeling the curing process in a finite element (FE) surroundings will probably give the most efficient tool to study the effectiveness of design choices concerning the geometry or the curing parameters. As an example of such a study, in [48] a (non-FE) model is used to study the effect of different cure temperature patterns on the hydrostatic, curing induced residual stress.

Having an accurate process model, higher reliability and dimensional accuracy could then be obtained in less iterative design cycles, leading to a reduction in waste and im-proved cost-effectiveness. The complexity of the complete modeling process will become clear by a listing of the required ingredients:

1. A cure progress model, to predict the degree of cure versus curing time, dependent on the resin temperature. Of course the same information can be obtained from a cure kinetics model, describing the evolution of the cure rate. NB degree of cure is also referred to as conversion;

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3. A cure dependent FE thermo-mechanical material model, to account for the devel-opment of the cure dependent mechanical properties;

4. A cure dependent initial strain model, to predict the thermal and chemical initial strains.

All of these components require a substantial number of experiments. The present work focuses on the cure dependent thermo-mechanical material model, mainly because of the two following deficiencies observed in literature.

The first is due to the fact that the material behaves viscoelastic, a phenomenon briefly introduced in section 2.2. The interference of the time dependencies due to viscoelasticity and chemistry prevents a conventional thermo-viscoelastic characterization technique (see section 2.3) during cure [10, 36]. Therefore, usually the thermo-viscoelastic properties are determined or estimated at (a number of) discrete degrees of cure. However, this determination or estimation of the development of the thermo-viscoelastic properties is usually rather poor. The shortcomings of the four approaches applied are mentioned below.

1. The assumptions made to limit the thermo-mechanical experiments during cure, often in the form of Time-Cure Superposition (see section 3.3.1), usually are not validated;

2. Experiments carried out at sub-Tg temperatures (see section 2.1.1 for a description of Tg), in which the curing reaction is thermally quenched, give no information on the thermo-viscoelastic properties at the cure temperature [44];

3. In [46] chemically different, fully cured materials are used to mimic the actual material in a partly cured state. The extent to which results from experiments carried out on these materials are representative of the actual material’s properties is unknown;

4. Experiments simply do not result in the full characterization of the necessary vis-coelastic properties [32].

The four approaches are discussed in section 3.3. The characterization method de-veloped in the present work does not suffer from these problems and gives one of the most direct estimates of the viscoelastic properties at arbitrary degree of cure and at the curing temperature.

Usually, the thermo-viscoelastic properties at various, but fixed, degrees of cure are captured in some model. Such model thus describes the viscoelastic properties or pro-cesses at arbitrary degree of cure, but these viscoelastic propro-cesses, e.g. relaxation, are not affected by ongoing cure.

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presented, however, not based on an experimentally obtained fixed-degree of cure model; some of the effects observed in the present experiments still are not included.

The present work also presents a theoretical framework to relate the ongoing cure relaxation function to the - whether or not experimentally obtained - fixed-degree of cure model.

Thus, more specifically, the present work aims at the development of a combination of thermo-viscoelastic characterization method and an accurate material model.

Rightly some researchers discuss the necessity to implement the viscoelastic effects, e.g. [6]. Many industrial curing processes take place at temperatures (far) above the ultimate glass transition temperature Tg∞, see section 2.1.1, of the material. For these processes, viscoelastic effects would be seen only for deformation rates much higher than those appearing due to the curing process; for the deformation rate of curing induced shrinkage the material behaves essentially elastic. Examples of elastic residual stress build-up calculations are given for example in [6, 35, 47].

For curing processes at temperatures below or around Tg∞, the actual (conversion dependent) Tg may get close to Tcure and then viscoelastic effects generally cannot be neglected. Curing processes at temperatures below or around Tg∞ are frequently ap-plied, for example in curing the matrix of large FRP products, in curing dental materials (acrylates) [40] or in molding processes in the electronics industry, because of the limited temperature resistivity of the components involved.

To be able to leave the heat transfer model out of the present investigations, some conditions have to be made with respect to the test specimen to be used. Generally, depending on the size of the specimen, the thermal boundary conditions and the heat generated during the exothermal reaction, the temperature inhomogeneity cannot be neglected and heat flow calculations have to be made. Cure dependent heat flow models are discussed for example in [60]. To be able to use the thermo-mechanical model without the heat transfer model the sample temperature, and with that the development of degree of cure, has to be homogeneous throughout the sample. Of course, this approximation is only valid for relatively thin samples, which is the pursuit in the present work.

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The structure of this thesis is as follows. Chapter 2 recapitulates and extends some textbook information which is useful in the remainder of this work. The term ‘polymers’, linear viscoelasticity and the matching mechanical characterization techniques come up for discussion.

The thermo-viscoelastic characterization principles, both existing and newly devel-oped, and constitutive equations for curing thermosets are discussed in Chapter 3.

In Chapter 4 the selection of a suitable measure for the degree of cure is discussed, as well as the cure kinetics, actually the cure progress, of an isothermal curing process to be used in the validation experiments.

Experimental details can be found in Chapter 5, together with specific information on the data processing applied. In this chapter also specific information on type and preparation of the thermosetting resin used can be found. All experiments in the present work were carried out on (model) epoxy resins, selected because of their similarity with commercial molding compounds as applied in the electronics industry. However, exper-imental methodology and modeling considerations are not expected to be valid for this specific thermosetting system alone. Chapter 5 also briefly describes chemical shrinkage measurements.

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Chapter 2 Recapitulation

In this chapter some literature and background information, useful in the remainder of this work, will be presented. In section 2.1 polymers in general are discussed. Since the thermosetting polymers discussed in the present work are viscoelastic, viscoelasticity will be introduced in section 2.2. Viscoelastic characterization techniques are finally discussed in section 2.3.

2.1 Polymers

According to [61] a polymer is a substance composed of macromolecules, where the latter are long sequences of covalently bonded building blocks, the monomers. The macro-molecules may be linear polymeric chains, branched polymeric chains or polymeric net-works. This variety in polymer structure is reflected in the variety in polymer properties. A possible classification of polymers is the one distinguishing thermoplastics and ther-mosets. Thermoplastics are composed of linear or branched polymeric chains which will melt upon heating and dissolve in suitable solvents. Thermoplastics therefore can be processed, e.g. molded, after production. In thermosets the covalently bonded 3D poly-meric network prevents both melting and dissolving, although swelling of the network may occur. Polymers may be amorphous, as network polymers are, or (semi-)crystalline. An important phenomenon in amorphous polymers is that they possess a so-called glass transition (temperature), which nature and implications will be discussed in the next section. Section 2.1.2 gives some brief information on the so-called polymerization process, i.e. the monomer to polymer conversion, as appearing during production and curing.

2.1.1 The Glass Transition Temperature

At sufficiently high temperatures, all polymers are either in a liquid or in a rubbery state. Cooling will show the divergency between crystalline and amorphous polymers. Crystalline polymers will crystallize at the melting temperature Tm, a transition that involves a stepwise change in the specific volume, see figure 2.1.1.

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usually abbreviated by Tg. The transition is characterized by a huge change in the material’s shear modulus, by highly viscoelastic behavior and, instead of the stepwise change in specific volume at Tm as observed in crystalline materials, a stepwise change in the slope of the specific volume versus temperature curve, and thus in the coefficient of thermal expansion. Crystalline Amorphous T g,1 T g,2 Temperature Specific volume

Figure 2.1.1: Specific volume versus temperature for a crystalline and an amorphous polymer. Tg,1 is obtained during an experiment with a high cooling rate, Tg,2 with a lower cooling rate.

The glassy state is a ‘non-equilibrium state’, determined by the material’s thermal history. The non-equilibrium state is reflected by the observed cooling rate dependency of Tg; the lower the cooling rate, the lower the Tg found, see also figure 2.1.1. There is still debate whether a limiting value for Tg would be reached for sufficiently low cooling rates.

Cooling an amorphous polymer into the glassy state will result in a high specific volume relative to that of a crystalline material. However, this relatively high specific volume is observed to decrease slowly towards the level of crystalline materials, an effect called volume retardation or aging. Volume retardation is very important for the long term creep behavior of polymers.

Dependent on the service temperature and Tg, amorphous polymers behave

1. glassy (T / Tg− 20 [◦_{C]), characterized by essentially elastic behavior and a high} shear modulus, order of magnitude 1000 [MPa];

2. viscoelastic (T ≈ Tg ± 20 [◦C]), characterized by highly viscoelastic behavior and an intermediate modulus;

3. and for T ' Tg+ 20 [◦C] either

(a) rubbery, in case of a network polymer; characterized by again essentially elastic behavior but a low modulus, order of magnitude 10 [MPa];

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Important in the remainder of this work are the observations that

1. For many network-forming polymers a one-to-one relationship between Tg and con-version is observed [50];

2. In the glass-rubber transition, the material behaves highly viscoelastic;

3. In passing-through the glass-rubber transition, there is a huge change in shear modulus.

2.1.2 Production and/or Curing

The building blocks of polymers are the so-called monomers. This section gives a brief de-scription of the polymerization process, i.e. the process to make polymers from monomers, and the resulting polymer structures. There is no chemical distinction between the poly-merization processes of thermoplastic and thermosetting materials. One just has to note that for thermoplastics, as well as the prepolymers for thermosetting resins, this poly-merization usually takes place in bulk at the production plant. Thermosetting resins, on the contrary, are in fact semi-manufactured bulk goods and specific, weighed out volumes are finally polymerized in or around another product, for example a woven fabric.

Initially the monomers, or prepolymers, are part of a mixture which often also con-tains a catalyst and sometimes a solvent. The monomers start to react with each other; polymer chains are formed. This is the so-called polymerization or, often used in case of thermosetting resins, curing process. It is controlled by temperature and the amount and type of catalyst.

One distinguishes between step polymerizations and chain polymerizations. In step polymerization, the polymer chain grows stepwise by reactions that can occur between any two molecular species available, i.e. momomer-monomer, monomer-polymer and polymer-polymer. In chain polymerization, the chain grows by reaction of monomers with a reactive end group on the polymeric chains. For the creation of such reactive end group an initiator is required. Epoxy-amine and iso-cyanate systems are examples of step-wise reacting systems, acrylates, for example, react chain-wise [36].

Furthermore, one distinguishes between polycondensation and polyaddition reactions. The first type of reactions involves the elimination of small molecules like H2O and HCl,

whereas in the second type the repeat units of the polymer have molecular formulae identi-cal to those of the monomers. Polyamines are prepared for example by polycondensation, epoxies are prepared by polyaddition reactions.

What is then the distinction between thermoplastic and thermosetting polymers? In thermosetting materials, opposite to thermoplastic materials, the functionality f0_{, i.e. the} number of reactive groups, of at least one of the reactants is larger than two. Whereas in thermoplastic materials linear polymer chains are formed, in thermosetting materials curing leads to the formation of branched polymer structures. These networks are often characterized by the so-called junction point or crosslink density [50].

During curing of a thermosetting resin gelation (probably) occurs, i.e. one of the

network molecules formed reaches the size of the reaction volume. This molecule is often

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2.2 Viscoelasticity

Viscoelastic materials behave either viscous or elastic, or somewhere in between, depen-dent on the time scale of the experiment. Section 2.2.1 starts with a one-dimensional (1D) introduction, which is generalized to 3D constitutive equations in section 2.2.2.

2.2.1 1D Introduction

For short loading durations viscoelastic materials behave mainly elastic. The one-dimensional constitutive equation for an ideal linear elastic material is Hooke’s law

σ = Ee² (2.2.1)

where σ represents the stress, Ee the elastic modulus and ² the strain. An ideal spring would respond similarly. For long loading durations viscoelastic materials behave (partly) viscous. The 1D constitutive equation for a viscous material is Newton’s law

σ = η ˙² (2.2.2)

where η represents the viscosity and ˙² = d

dt² the strain rate. A dashpot would respond similarly. A viscoelastic material combines both effects; actually stress does not depend on either the strain or the strain rate alone, but on the entire strain history and thereby becomes time dependent. The same holds for the strain dependence on applied stresses. Thinking in the mechanically analog models of springs and dashpots, one of the most simple constitutive equations for viscoelastic materials is one describing the response of a series connection of one spring (² = ²1) and one dashpot (² = ²2), the so-called Maxwell

model ˙² = ˙²1+ ˙²2 = 1 Ee ˙σ + 1 ησ (2.2.3)

See also figure 2.2.1a. Let’s calculate the response of this material to a step strain, applied

Figure 2.2.1: Mechanical analogs of viscoelastic materials. a) Maxwell element and b) Kelvin element.

at t = ξ0, by applying Laplace transformation. The step strain is represented by

² (t) = ²0∆ (θ) , where ∆ (θ) =

½

0, θ < 0

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−4 −2 0 2 4 0 0.2 0.4 0.6 0.8 1 log(θ) σ ( θ ) τ 0 20 40 60 0 0.2 0.4 0.6 0.8 1 θ σ ( θ ) τ

Figure 2.2.2: Exponential decay of stress upon stepwise application of strain, according to equation 2.2.7.

where θ = t − ξ0. The Laplace transform of a function f (t), i.e. L (f (t)), is represented

by f (s), where where s represents the Laplace transform variable. The Laplace operator

L is defined as [26] L (f (t)) = ∞ Z t=0− f (t) e−st_dt _(2.2.5)

It is not difficult to show that the Laplace transform of the derivative of f (t), i.e. ˙f (s) = −f (0−_{) + sf (s), where f (0}−_{) is the initial value. The Laplace transform of equation} 2.2.3 can now be written as

²0 = sσ Ee +σ η → σ = ²0Ee s +Ee η (2.2.6) Transformation to the time domain learns, with L−1¡ 1

α+s ¢ = e−αt _[13]; σ (θ) = Eeexp µ −Ee η θ ¶ ²0 = Eeexp µ −Ee η (t − ξ0) ¶ ²0 (2.2.7)

This stress decay is represented schematically in figure 2.2.2 for ²0 = Ee = 1. Here

the time dependent modulus is the so-called relaxation modulus and is usually written in the form E (θ) = Eeexp µ −θ τ ¶ (2.2.8) In this case τ = η/Ee, i.e. that time the modulus has decreased by a factor e−1 _{≈ 0.3679,} is usually called the relaxation time.

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−40 −3 −2 −1 0 1 2 3 4 1 2 3 4 5 6 log(θ) E( θ )

Figure 2.2.3: Schematic ‘experimental’ relaxation modulus (solid line) and contributions of discrete model relaxations by equation 2.2.9 (dashed lines).

practice, the experimentally found relaxation modulus, for example the solid line in fig-ure 2.2.3, is often approximated using a number of discrete terms in the form of equation 2.2.8, which are referred to in the present work as model relaxations, i.e.

E (θ) ≈ N X n=1 Enexp µ − θ τn ¶ (2.2.9) The contributions of the discrete model relaxations are represented by the dashed lines in figure 2.2.3. Equation 2.2.9 is often referred to as Prony series and it describes the step relaxation response of a series of shunted Maxwell elements, the so-called generalized Maxwell model. Since the modulus of network polymers often also has an elastic part, a typical relaxation function for these materials could be

E (θ) ≈ E∞+ N X n=1 Enexp µ − θ τn ¶ (2.2.10) The elastic part is denoted by E∞ since it reflects the remainder of the modulus at infinite time. Next to E∞, Erubber is also used since this remaining modulus is often

attributed to rubber elastic behavior of the polymeric network.

In the present work, linear viscoelasticity is assumed, i.e. the relaxation modulus does not depend on either stress or strain level. If the material behaves linear, the total stress response due to different load steps is the linear superposition of the response to each of these load steps

σ (t) =

I X

i=1

E (θi) ²i∆ (θi) , θi = t − ξi (2.2.11)

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formu-lation σ (t) = t Z ξ=t0 E (t − ξ) ∂² ∂ξdξ (2.2.12)

where t0 the ‘initial stress-free time’. If stress is chosen as the controlling parameter, the

corresponding creep formulation of equation 2.2.12 is obtained

² (t) = t Z ξ=t0 J (t − ξ)∂σ ∂ξdξ (2.2.13)

where J (t − ξ) is the so-called creep compliance. In this case t0 is the ‘initial strain-free

time’. Equations 2.2.12 and 2.2.13 are called ”hereditary integrals” [13] and they clearly reflect the strain and stress history dependence of stress and strain, respectively.

To find the relation between E (t − ξ) and J (t − ξ), assume the material behavior can be described by a model build up from some combination of Maxwell and Kelvin elements, where the latter consists of a dashpot and a spring in parallel, see also figure 2.2.1b. According to [13] the differential equation of such a model has the form

p0σ + p1˙σ + p2σ + · · · = q¨ 0² + q1˙² + q2¨² + . . . (2.2.14) or shortly K X k=0 pk dk_σ dtk = L X l=0 ql dl_² dtl (2.2.15)

Under the condition that ² (t) and all its time derivatives vanish for t = 0−_{, the Laplace} transform of this equation can be written as

P (s) σ = Q (s) ² (2.2.16) where P (s) = PK k=0 pksk and Q (s) = L P l=0

qlsl. For the Laplace transform of the creep compliance, i.e. the strain response on σ = ∆ (t), one can then write

J (s) = P (s)

sQ (s) (2.2.17)

Similarly for the relaxation modulus, which is the response of equation 2.2.15 on ² = ∆ (t)

E (s) = Q (s)

sP (s) (2.2.18)

Comparison of equations 2.2.18 and 2.2.17 learns

J (s) E (s) = s−2 _(2.2.19)

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the constitutive equation to be used in that case is obtained by substitution of equation 2.2.10 into equation 2.2.12. Assuming ² (t0) = 0 one obtains

σ (t) = E∞² (t) + N X n=1 t Z ξ=t0 Enexp µ −t − ξ τn ¶ ∂² ∂ξdξ, t ≥ t0 (2.2.20)

2.2.2 3D Viscoelasticity

The 3D behavior of linear elastic, isotropic materials is characterized by just two material constants [7]. These materials are fully characterized by the equations

σd_{= 2G²}d _(2.2.21)

and

σh = 3K²v (2.2.22)

Here σd _{and ²}d _{are the deviatoric parts of the stress and strain tensor, respectively. The} tensors σh _{and ²}v _{represent the hydrostatic part of the stress tensor and the volumetric} part of the strain tensor, respectively. The material constants G and K are named the shear and the bulk modulus. Some definitions

σ = σh_{+ σ}d ² = ²v_{+ ²}d σh ₌ 1 3tr (σ) I ²v ₌ 1 3tr (²) I (2.2.23)

Here I is the unit tensor. In terms of Cartesian tensor components one can write for the traces

tr (σ) = σii= σ11+ σ22+ σ33

tr (²) = ²ii= ²11+ ²22+ ²33 (2.2.24)

With the tensor components written in vector form, equation 2.2.21 could be written as         σd 11 σd 22 σd 33 σd 12 σd 23 σd 31         = 2G                 1 1 0 1 1 0 1 1         − 1 3         1 1 1 1 1 1 0 1 1 1 0 0                         ²11 ²22 ²33 ²12 ²23 ²31         (2.2.25)

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The total stress is the summation of the hydrostatic and deviatoric parts and thus one could have written also

        σ11 σ22 σ33 σ12 σ23 σ31         =         K + 4 3G K − 2 3G K − 2 3G 0 0 0 K +4 3G K − 23G 0 0 0 K + 4 3G 0 0 0 symm. 2G 0 0 2G 0 2G                 ²11 ²22 ²33 ²12 ²23 ²31         (2.2.27) or with G = E 2 (1 + ν), K = E 3 (1 − 2ν) (2.2.28)

where ν is Poisson’s ratio, in the form         σ11 σ22 σ33 σ12 σ23 σ31         = E (1 − ν) (1 + ν) (1 − 2ν)         1 ν 1−ν 1−νν 0 0 0 1 ν 1−ν 0 0 0 1 0 0 0 symm. 1−2ν 1−ν 0 0 1−2ν 1−ν 0 1−2ν 1−ν                 ²11 ²22 ²33 ²12 ²23 ²31         (2.2.29)

The linear viscoelastic equivalents of equations 2.2.21 and 2.2.22 are, similar to equation 2.2.12 σd(t) = 2 t Z ξ=t0 G (t − ξ)∂² d ∂ξ dξ (2.2.30) and σh(t) = 3 t Z ξ=t0 K (t − ξ)∂² v ∂ξ dξ (2.2.31)

for the deviatoric and volumetric parts, respectively. Again in matrix notation         σ11(t) σ22(t) σ33(t) σ12(t) σ23(t) σ31(t)         = t Z ξ=t0         K +4 3G K − 23G K − 23G 0 0 0 K + 4 3G K − 23G 0 0 0 K + 4 3G 0 0 0 symm. 2G 0 0 2G 0 2G         ∂ ∂ξ         ²11 ²22 ²33 ²12 ²23 ²31         dξ (2.2.32) where, of course, K = K (t − ξ) and G = G (t − ξ). See also equation (11.2-9) in [53].

As far as in the present work non-curing viscoelastic behavior is concerned, equations of the type of equation 2.2.10 will be used for the relaxation functions

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K (θ) = K∞+ P X p=1 Kpexp µ −θ τp ¶ (2.2.34) Here, Kp and Gn are called relaxation strengths and the τp and τn are called relaxation times. For the next chapters it is useful to have a description of the step relaxation shear response. For σ12 one has from equation 2.2.32

σ12(t) = 2 t Z ξ=t0 G (t − ξ)∂²12 ∂ξ dξ

In case of step relaxation experiment, ²12(ξ) = ²0,12∆ (ξ0), where ∆ (ξ0) is defined as in

equation 2.2.4. Then one obtains for

∂²12 ∂ξ = ²0,12δ 0_(ξ 0) → δ0(ξ0) = ½ 0, ξ 6= ξ0 ∞, ξ = ξ0 , ∞ Z ξ=−∞ δ0(ξ0) dξ ≡ 1 Then σ12(t) = 2²0,12 t Z ξ=t0 G (t − ξ) δ0(ξ0) dξ = 2²0,12 t Z ξ=t0 G (t − ξ0) δ0(ξ0) dξ = 2G (t − ξ0) ²0,12 t Z ξ=t0 δ0(ξ0) dξ ≡ 2G (t − ξ0) ²0,12, t0 ≤ ξ0 ≤ t

Substitution of equation 2.2.33 for G (t − ξ0) gives with θ = t − ξ0

σ12(θ) = 2 " G∞+ N X n=1 Gnexp µ −θ τn ¶# ²0,12 (2.2.35)

2.3 Characterization of Viscoelastic Materials

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2.3.1 Creep and Relaxation Experiments

In creep and relaxation experiments a transient response, characteristic of viscoelastic materials, is measured. In creep experiments the time dependent strain upon application of a constant stress is measured. In relaxation experimens the time dependent stress upon the application of a constant strain is measured. For unambiguous interpretation the loading or straining should be applied stepwise. In case of shear relaxation, for example, σ (t) /2²0 is then a direct measure for the relaxation modulus, see also equation

2.2.35.

Since stepwise loading or straining would require infinitely high loading or straining rates it is practically not feasible; short term responses therefore have to be looked at with some suspicion. Creep and relaxation tests are particularly useful in the case of non-linear viscoelasticity, where interpretation of the results from dynamic mechanical experiments is rather difficult [14]. In [57] several apparatus for transient experiments are presented. See also [12] for detailed experimental descriptions.

2.3.2 Dynamic Mechanical Analysis

In dynamic mechanical analysis (DMA) a specimen is subjected to sinusoidal, or har-monic, excitation. In the present work, the sinusoidal strain is prescribed. As starting point the stress component σ12 from equation 2.2.32 will be used, with equation 2.2.33

substituted for G (t) and t0 = 0:

σ12(t) = 2G∞²12(t) + 2 N X n=1 t Z ξ=0 Gnexp µ −t − ξ τn ¶ ∂²12 ∂ξ dξ (2.3.1)

under the assumption ²12(t = 0) = 0. Since equation 2.2.32 is for linear viscoelastic

materials, the same holds for the following derivation. Selecting only one Prony term and substitution of ²12= ²0sin ωξ gives

σ12,n(t) = 2ω t Z ξ=0 Gnexp µ −t − ξ τn ¶ ²0cos ωξdξ (2.3.2) = 2ωτnGnexp µ −t − ξ τn ¶ ²0cos ωξ ¯ ¯ ¯ ¯ t ξ=0 + 2ω2_τ n t Z ξ=0 Gnexp µ −t − ξ τn ¶ ²0sin ωξdξ

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Thus σ12,n(t) = 2ω t Z ξ=0 Gnexp µ −t − ξ τn ¶ ²0cos ωξdξ = 2ωτnGnexp µ −t − ξ τn ¶ ²0cos ωξ ¯ ¯ ¯ ¯ t ξ=0 + 2ω2_τ2 nGnexp µ −t − ξ τn ¶ ²0sin ωξ ¯ ¯ ¯ ¯ t ξ=0 − 2ω3_τ2 n t Z ξ=0 Gnexp µ −t − ξ τn ¶ ²0cos ωξdξ (2.3.5) = 2ωτnGnexp µ −t − ξ τn ¶ ²0cos ωξ ¯ ¯ ¯ ¯ t ξ=0 + 2ω2_τ2 nGnexp µ −t − ξ τn ¶ ²0sin ωξ ¯ ¯ ¯ ¯ t ξ=0 (2.3.6) − ω2τ_n2σ12,n(t) Rearranging leads to σ12,n(t) = 2Gn ωτn 1 + ω2_τ2 n ²0cos ωt + 2Gn ω2_τ2 n 1 + ω2_τ2 n ²0sin ωt − 2Gn ωτn 1 + ω2_τ2 n ²0exp µ − t τn ¶ (2.3.7) The stress response thus contains a transient term. After stabilization of the stress signal, to be discussed later on, one obtains for the stress

σ12(t) = 2G∞²0sin ωξ + N X n=1 σ12,n(t) (2.3.8) = 2 " G∞+ N X n=1 Gn ω2_τ2 n 1 + ω2_τ2 n # ²0sin ωt + 2 N X n=1 Gn ωτn 1 + ω2_τ2 n ²0cos ωt ≡ 2G0_² 0sin ωt + 2G00²0cos ωt

Here G0 _{and G}00 _{are called the dynamic moduli. More specifically}

G0 _{= G} ∞+ N X n=1 Gn ω2_τ2 n 1 + ω2_τ2 n ≡ G∞+ N X n=1 G0_| n(ω, τn) (2.3.9)

is the so-called storage modulus and

G00 = N X n=1 Gn ωτn 1 + ω2_τ2 n ≡ N X n=1 G00|_n(ω, τn) (2.3.10)

the so-called loss modulus. Note that the stress contribution with the storage modulus, in equation 2.3.8, is in-phase with the applied strain. Therefore the storage modulus is responsible for the elastic part of the stress. The stress contribution with the loss modulus is out-of-phase with strain and is therefore responsible for the viscous part of the stress. Experimentally, the (sinusoidal) strain may be applied and the (sinusoidal) stress, which is shifted in time, is measured

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Comparison with equation 2.3.8 learns G0 ₌ σ0 2²0 cos δ, G00 ₌ σ0 2²0 sin δ (2.3.12)

Equilibration of DMA response Experimentally, it is thus important that the ma-terial is under mechanical excitation for a number of cycles larger than those required for stabilization of the transient effects, let’s call these the equilibration cycles. The following analysis will show the number of equilibration cycles required.

To be conservative, the behavior of just one of the N terms of equation 2.3.7 is elaborated. Suppose the transient variation relative to the stress sine amplitude, over a number of excitation cycles c1 during which DMA data are measured for determination

of the dynamic moduli, should not exceed some tolerance ζ ¿ 1, i.e.

σt(t) − σt ¡ t +2πc1 ω ¢ σ0 ≤ ζ (2.3.13)

where σt(t) is given by the last term of equation 2.3.7. From equation 2.3.7 the stress amplitude can be derived

σ0 = 2Gn²0 ωτn (1 + ω2_τ2 n) 1 2 (2.3.14) With this the criterion could be written as

exp µ − t τn ¶ · 1 − exp µ −2πc1 ωτn ¶¸ ≤¡1 + ω2τ_n2¢12 _ζ _(2.3.15)

The time one should apply excitation before actually measuring the response is

t ≥ −τnln (1 + ω2_τ2 n) 1 2 ζ 1 − exp ³ −2πc1 ωτn ´ (2.3.16)

The number of excitation periods c2 before actually measuring the response is then

c2 ≥ − ωτn 2π ln (1 + ω2_τ2 n) 1 2 _ζ 1 − exp ³ −2πc1 ωτn ´ (2.3.17)

This number is plotted for different values of ωτnin figure 2.3.1. It is somewhat surprising that c2 becomes negative for certain values of ωτn. This is due to the fact that for these values of ωτncondition (equation) 2.3.13 is already met at the start of the excitation; the transient variation is simply negligible compared to the stress amplitude. The negative parts of the curves shown in figure 2.3.1 therefore have to be ignored.

Of course, also the transient effects of former harmonic excitations should be more or less stabilized within the time span of the new excitation. Suppose that for this excitation the frequency was ωp and the initial transient terms are stabilized. From equation 2.3.5 the residual stress after a whole number of excitation periods can be derived as

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−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −5 0 5 a b log(ωτ n) c 2 a b

Figure 2.3.1: Number of equilibrium cycles required, a) in case ζ = 0.05 and c1 = 1 and

b) in case ζ = 0.01 and c1 = 3.

Here t000 _{is the time after the excitation at ω}

p has stopped. Except the ωp, t000 and the sign this equation for the residual stress is similar to that for the transient term in equation 2.3.7. If one assumes that ωp is smaller or at least not orders of magnitude larger than ω, the current excitation frequency, the residual stress of the former excitation may partly cancel out the transient term of the current excitation.

At least one can conclude that generally all transient effects active during the current excitation are diminished sufficiently after some five excitation periods.

Viscoelastic Characterization using DMA Single frequency DMA is often used to track the development of ‘the’ modulus during cure. This knowledge is mainly useful for decision-making in polymer processing. Naturally, multi-frequency DMA is interesting to study the frequency dependency of the modulus, as is useful if one deals with vibrations. Moreover, from these multi-frequency DMA results also the relaxation modulus G (θ), as in figure 2.2.3, can be approximated.

Somehow this could be expected; within the constant periodic time of the harmonic excitation in DMA experiments, i.e. 2πω−1_{[s], parts of the relaxation curve at (relaxation)} times θ ¿ 2πω−1_{, or in model terminology: model relaxations with characteristic times}

τn ¿ 2πω−1, will not play a role in the dynamic modulus measured. Parts or terms having characteristic times much larger than 2πω−1 _{will contribute entirely elastic to the} dynamic modulus. In terms of model relaxations: just that terms with τn ≈ 2πω−1 are expected to contribute viscoelastically to the dynamic modulus.

By taking a closer look at the shear DMA response of a viscoelastic material, char-acterized by a discrete model relaxation approximation in the form of equation 2.2.33, a more detailed relation between the time and frequency dependent response of viscoelastic materials is finally obtained. Stepwise:

• In figure 2.3.2 the contributions of all N model relaxations from equation 2.2.33,

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com-pared. In this case, all Gn, n = 1 . . . N are chosen equal and the relaxation times −4 −2 0 2 4 0 1 2 3 4 5 6 k k+1 k+2 k+3 k+4 log(θ) G( θ ) −4 −2 0 2 4 0 1 2 3 4 5 6 k k+1 k+2 k+3 k+4 log(ω) G’( ω )

Figure 2.3.2: Comparison between contributions of discrete model relaxations to relax-ation modulus and storage modulus, respectively.

are logarithmically equally spaced. It is observed here that each model relaxation represents a limited part of both the relaxation modulus and the storage modulus, as seen on the log (θ) and the log (ω) axis, respectively.

• In setting up DMA experiments, to approximate G (θ), it can be important to

know where a model relaxation with τ = τn shows up on the frequency axis. For this, one has to take a closer look at the functions G0_|

n(ω, τn) and G00|n(ω, τn) in the equations 2.3.9 and 2.3.10. In figure 2.3.3 a graphical representation of these functions is given. Just at frequencies around ωτn = 1 highly viscoelastic effects

−20 −1.5 −1 −0.5 0 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 log(ωτ n) G’| n ( ω , τ n ),G"| n ( ω , τ n ) G"| n G’| n

Figure 2.3.3: The functions G0_|

n(ω, τn) and G00|n(ω, τn) from equations 2.3.9 and 2.3.10.

are observed; characteristic is that G00_|

n is comparable to G0|n and that both G00|n and the slope in G0_|

n show a maximum. For this reason

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could be seen as an activation condition for the (model) relaxation with character-istic time τn.

It is concluded here that, if the relaxation function is approximated by equation 2.2.33, then primarily the effects of the specific nth _{model relaxation are probed by} frequency variation around its activation frequency ωact

n = τn−1.

• With this knowledge, an appropriate experimental frequency range can be

esti-mated. Afterwards the experimental dynamic moduli data then often have to be converted to relaxation modulus data. One way of relating these data is fitting the measured G0_{(ω) and G}00_{(ω) by the equations 2.3.9 and 2.3.10 and use the found} parameters in equation 2.2.33. This method may work well if the full dynamic modulus curve, i.e. including rubber and glassy levels, has to be transformed. However, in case a storage modulus curve for limited frequency range has to be converted, some problem will arise. This kind of curves would be fitted using equa-tions 2.3.9 and 2.3.10, such that the fit represents the experimental data. Figure 2.3.3 shows that model relaxations are effective within roughly 1 or 2 frequency decades. Depending on the chosen spacing of relaxation times, the fitting function usually will generate artificial storage modulus data outside the experimental fre-quency range. This means that when the parameters of equations 2.3.9 and 2.3.10, as found by the fitting procedure, are used in equation 2.2.33 to predict the relax-ation modulus, at least parts of both the short and long term response should be considered as unreliable.

• Therefore it is desirable to relate the response at discrete frequencies ωi in the DMA experiments to discrete times θi in the relaxation experiments

G (θi) = f (G0(ωi) , G00(ωi))

Some simple relations found in literature relate the relaxation modulus to the stor-age modulus G (θ) = G0_(ω)| ω=2/πθ [53] G (θ) = G0_(ω)| ω=1/θ [12, 36] (2.3.20) In the present work a similar relation will be derived, based on the observation that G (θ) and G0_{(ω) from figure 2.3.2 seem to be reflected images of one another.} If one would replace the labels of both x-axes in figure 2.3.2 by x1, the following

‘symmetry’ is observed

G (x1− C) = G0(− (x1 − C)) (2.3.21)

where C is the x1-coordinate of the symmetry axis. Rewriting gives

G (x1− C) = G0(−x1+ C)

G (x1) = G0(−x1+ 2C) (2.3.22)

By plotting both G (x1) and G0(−x1+ 2C) versus x1 there is some value of C for

which the resulting G0_{-curve lays on top of G (x}

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constant C appears to be −0.1259, see also figure 2.3.4. In this figure x1 = log θ is

plotted, since G (x1) is used as reference curve. In case of G0, x1 = log ω, and thus

G (log θ) = G0_{(− log ω + 2 log c)}

G (θ) = G0¡_c2_ω−1¢ With C = log c one obtains c2 _{= 0.56 and thus}

G (θ) = G0_(ω)|

ω=0.56θ−1 (2.3.23)

In figure 2.3.4 also the results of the equations 2.3.20 are plotted.

−40 −3 −2 −1 0 1 2 3 4 1 2 3 4 5 6 G(θ) x₁=log(θ) G(x 1 ), G’(−x 1 +2C)) ω=1/θ ω=2/πθ G’ ω=0.56/θ

Figure 2.3.4: Relaxation modulus G (θ) according to equation 2.2.33 and storage modulus

G0_(θ)|

θ=f (ω) according to equation 2.3.9 (N = 51).

Equation 2.3.23 has an approximate character, since it is based on the apparent symmetry in the time and frequency response of model functions. Nevertheless, the relation appears quite useful in the present work. One could say that, to some extent, single frequency DMA probes the G (θ) at discrete (relaxation) times θ. Practical Limitations and the Effect of Temperature Theoretically, a sufficiently large frequency window allows the effect of all relaxations to be measured, or in model terms: will activate all model relaxations. However, for both practical and experimental reasons the frequency range is limited and only a part of the relaxation effects will be measured. In figure 2.3.5a this is illustrated for some specific temperature T1. In this

figure the experimental frequency window limits are represented by the vertical lines. The solid line within this experimental frequency window represents the (normalized) storage modulus, which is usually measured. The inaccessable parts of the viscoelastic properties are presented by the dotted lines.

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−30 −2 −1 0 1 2 3 0.5 1 T 1 T2 a G’( ω ,T) −30 −2 −1 0 1 2 3 0.5 1 n=k k+1 k+2 k+3 k+4 k+5 G’| n ( ω ,T) b −30 −2 −1 0 1 2 3 0.5 1 c n=k+2 k+3 k+4 k+5 k+6 k+7 log(ω) G’| n ( ω ,T)

Figure 2.3.5: Effect of temperature. a) Schematic experimental storage modulus at T1

and T2 > T1 (solid line) , b) model relaxations active within experimental frequency

window at T1 , c) model relaxations active within experimental frequency window at T2

(dashed lines). Inaccessible parts are presented as dotted lines.

able to measure viscoelastic effects which are at ambient or operation temperature far be-yond experimental reach. Suppose just the relaxation times are temperature dependent. For the model equation, this implies that the most general introduction of temperature dependence in the equations 2.3.9 and 2.3.10, i.e.

G0_{(T ) = G} ∞(T ) + N X n=1 Gn(T ) ω2_τ2 n(T ) 1 + ω2_τ2 n(T ) , G00_{(T ) =} N X n=1 Gn(T ) ωτn(T ) 1 + ω2_τ2 n(T ) (2.3.24) would reduce to G0_{(T ) = G} ∞+ N X n=1 Gn ω2_τ2 n(T ) 1 + ω2_τ2 n(T ) , G00_{(T ) =} N X n=1 Gn ωτn(T ) 1 + ω2_τ2 n(T ) (2.3.25)

As will be seen in Chapter 5, both the rubbery modulus G∞ and the relaxation

strengths Gnactually are temperature dependent. In practice, however, the temperature dependence of the relaxation strengths is often neglected. This approximation will also be adopted in the present work since for the viscoelastic characteristics of the material the effect of temperature on the relaxation times is much larger than on the relaxation strengths. Also the temperature dependence of the rubbery modulus will be neglected.

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be active in the experimental frequency window than at T = T1. This is indicated in

figure 2.3.5b and 2.3.5c. However, the variables in equation 2.3.25 are yet unknown. The curve G0_{(ω, T}

2), however, can be determined experimentally. Now, somehow each curve

G0_{(ω, T}

i) , i = 1 . . . I has to be related to a number of model relaxations from equation 2.3.25. These modeling steps, which are not evident, generally make use of the principle of time-temperature superposition (TTS), as will be discussed in the next section.

2.3.3 Time-Temperature Superposition

Thermo-viscoelastic characterization usually results in a set of relaxation or dynamic modulus curves, obtained at different temperatures. To convert this set of data to another, showing all viscoelastic effects at one, arbitrary, temperature, the principle of Time-Temperature Superposition is usually invoked. The principle states that all relaxation times, thus also the model relaxation times, respond similarly to changes in temperature

τn(T ) = a (T, T0) τn(T0) , n = 1 . . . N (2.3.26)

Materials for which this statement is valid are called thermo-rheologically simple ma-terials. If only the relaxation times are temperature dependent, time-temperature su-perposition implies that DMA-results obtained at different temperatures can be shifted along the log (ω)-axis, thereby virtually extending the frequency window of the experi-ments. The results of such a shifting procedure for the storage modulus are presented schematically in figure 2.3.6, together with the experimental data in the frequency range

ω = 100.6_{. . . 10}1_{. In fact some overlap between data obtained at adjacent temperatures}

is desirable for sufficient accuracy. The shift factors follow from this shifting procedure, which can be performed optically or numerically. The shift factors a (T, T0) found give

−8 −6 −4 −2 0 2 log(ω) [rad s−1] G’( ω ,T) T=T 1 T=T 2 T=T 3 T=T 4 T 1>T2>T3>T4 G glass G_∞

Figure 2.3.6: Time-temperature superposition. Schematic storage modulus data, ‘mea-sured’ at ω = 100.6_{. . . 10}1_{, are shifted such to form a master curve. In this schematic figure}

the lowest temperature was taken as reference temperature, i.e. data for this temperature is not shifted.

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DMA-results shifting is usually performed using tan δ = G00_/G0 _{instead of G}0 _{or G}00_{, since} the glassy and rubbery plateau values of the moduli, denoted as Gglass and G∞ in figure 2.3.6 respectively, actually are also temperature dependent. Finally, the shifted G0 _{or G}00 data are fitted using either equation 2.3.9 or 2.3.10, to give the relaxation strengths Gn, or the relaxations times τn, or both. Thereby the thermo-viscoelastic characterization is complete: the obtained parameters Gn, τn and a (T, T0) can be used for example in

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Chapter 3 Viscoelasticity of Curing Thermosets

- Experiments and Modeling

The previous chapter discussed some standard modeling and characterization techniques for linear viscoelastic materials. These standard techniques are incapable for the mod-eling and characterization of curing thermosets, where the viscoelastic properties change drastically with ongoing cure. In the present chapter the standard techniques are adapted to be applicable for curing thermosets.

3.1 Preliminary Considerations

In this chapter the development of a cure dependent, linear viscoelastic constitutive model is discussed. The cure dependency is expressed by means of the parameter α, which stands for the degree of cure, or conversion. NB 0 ≤ α ≤ 1.

Unless mentioned otherwise, only the deviatoric part of the constitutive equations is presented in the present chapter. For the constitutive model developed here, the analogous volumetric part will be presented in section 3.4.5.

Linear viscoelastic constitutive equations can have the following form, cf. equation 2.2.30 σd_{(t) = 2} t Z ξ=t0 G (t − ξ)d² d dξdξ (3.1.1)

Formally, the constitutive model for linear viscoelastic materials with ongoing cure could be written in more or less the same form, i.e.

σd_{(t) = 2} t Z ξ=t0 Gc_{(t, ξ)}d²d dξdξ (3.1.2)

where Gc_{(t, ξ) is now the relaxation function with ongoing cure.}

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Preferably, Gc_{(t, ξ) should be measured directly, at least for isothermal curing} pro-cesses; by measuring Gc_{(t, ξi}_{), starting from a number of α (ξi) , i = 1 . . . I, a general} model Gc_{(t, ξ) would be easily made.}

However, for the modeling of arbitrary, non-isothermal curing processes, these relax-ation curves seem less useful. Moreover, generally direct relaxrelax-ation or creep experiments during cure are not possible due to the simultaneously appearing chemical shrinkage. Simple shear experiments may be an exception, since the applied and chemically induced deformation are more or less uncoupled here. I.e. for this specific case the relaxation function, and thereby the deviatoric part of the constitutive equation, i.e. eq. 3.1.2, the-oretically could be determined directly. This is not performed in the present work, since the aim is here to develop a method that is generally applicable, i.e. also useful for the determination of the bulk relaxation modulus. An additional reason that direct Gc_{(t, ξ)} measurements are not carried out in the present work are the experimental problems that are expected for relaxation measurements on curing specimen 1_.

An alternative way to construct Gc_{(t, ξ) thus has to be found.}

In literature several experimentally supported relaxation function models describing the viscoelastic properties at any, but constant, degree of cure are developed, i.e.

G (xi(αj) , θ) , θ = t − ξ (3.1.3)

where xirepresent some internal variables depending on the degree of cure αj, for example the relaxation times and relaxation strengths of equation 2.2.33. This model should be capable of reproducing for example the curves shown in figure 3.1.1. The observed curing effects are explained in section 3.3.5. Generally the resulting model cannot be directly applied for curing materials. For curing materials the xi(α) are also time dependent, by

α (t0_{) where t}0 _{is curing time. For the ongoing cure case, the relaxation function is some} function

Gc(xi(α (ξ ≤ t0 ≤ t)) , t, ξ) (3.1.4)

This function has to take into account the evolution of the xi(α (t0)) within the curing time interval ξ ≤ t0 _{≤ t and the effect of that evolution on the relaxation process. Although} this equation may ultimately be written as Gc_{(t, ξ), like in equation 3.1.2, the xi} _are mentioned explicitly because they form the necessary link to the observed viscoelastic behavior at several degrees of cure, as represented by equation 3.1.3.

In practice also equation 3.1.3 is sometimes used as relaxation function in ongoing cure situations. Kiasat [32] for example uses

Gp_(x

i(α (ξ)) , t − ξ) (3.1.5)

1_{In practice relaxation tests during cure are rather difficult. At low degrees of cure, samples are very}

(41)

−4 −2 0 2 4 0 0.5 1 1.5 2 2.5 3 log(θ) log(G( α , θ )) α₁ α2 α₃ −4 −2 0 2 4 0 200 400 600 800 1000 log(θ) G( α , θ ) α₁ α₂ α₃

Figure 3.1.1: Relaxation shear modulus for different degrees of cure, α1 < α2 < α3. NB

θ = t − ξ.

i.e. relaxation after load application is described by a relaxation function based on the degree of cure at that load application time; the effect of curing on the subsequent relaxation process is not taken into account.

Remark The effect of temperature on the relaxation function can be treated similarly to that of the degree of cure; simply replace the α in the equations 3.1.3 to 3.1.5 by α, T .

If a relaxation function of the type 3.1.3 is used in a constitutive model for the ongoing cure case, both the relaxation function and the constitutive model are referred to as partly

cure dependent (PCD). For that reason a superscript p is used in equation 3.1.5.

In contrast, fully cure dependent (FDC) relaxation functions will denote relaxation functions which take into account the effect of cure on the relaxation process itself, i.e. of the type of equation 3.1.4. For these functions, a superscript f will be used from now on, instead of the c in equation 3.1.4. Possible differences in response of the two relaxations functions are schematically shown in figure 3.1.2. The performance of the partly cure dependent models and the fully cure dependent models will be compared in Chapter 7.

Since the PCD relaxation function, equation 3.1.3, seems to be the only function that can be experimentally determined, the fully cure dependent constitutive modeling will have to be based on this function. Therefore the development of a PCD model will be considered as a first step in the development of an FCD constitutive model. The available and applied methods are discussed in section 3.3.

The development of the fully cure dependent relaxation function Gf_{(t, ξ) itself, using} expression 3.1.3 as starting point, is then considered as the second step. This step, which is usually missing in literature, is presented in section 3.4.

Chapter 4 describes the necessary cure kinetics, i.e. the development of α (t0_{) from} equation 3.1.4. Chapter 5 discusses the experiments to be done for the PCD model

Gp_(x

Mechanical characterization and modeling of curing thermosets