2002:40

DOCTORAL THESIS

Predictions of Manufacturing Induced Shape Distortions - high performance thermoset composites

J. Magnus Svanberg

Department of Applied Physics and Mechanical Engineering Division of Polymer Engineering

2002:40 • ISSN: 1402 - 1544 • ISRN: LTU - DT - - 02/40 - - SE Predictions of manufacturing induced shape distortions -high performance thermoset composites

J. Magnus Svanberg

SICOMP AB Box 271 SE-941 26 Piteå Sweden

and

Division of Polymer Engineering Department of Applied Physics and Mechanical Engineering Luleå University of Technology SE-971 87 Luleå Sweden ABSTRACT

High performance composites usually consist of continuous fibres and a thermoset matrix. A well-known example is carbon fibre epoxy composites. When this kind of material is cured residual stresses and/or shape distortions are produced owing to thermally and chemically induced volumetric strains. The cure means the manufacturing step where the thermoset matrix is transformed from a liquid to a solid material. It is a quite complex thermal- chemical- mechanical process that in addition to volumetric strains, involves heat generation and dramatic changes in mechanical properties. For manufacturing of parts with high shape tolerances, such as aircraft components, the geometry of the mould is compensated to accommodate for shape distortions. Today this is made based on thumb rules and experience followed by trials. This is time consuming and expensive. Development of a tool for prediction of shape distortions and residual stresses is therefore an important step towards more optimised manufacturing of composites. The present thesis, consisting of five papers, describes the development and validation of a simulation tool for prediction of shape distortion and residual stresses. In the first paper a typical material and manufacturing process for high performance composites was used to experimentally investigate the effects from the cure temperature on spring-in of angle sections. The experimental results were interpreted in terms of mechanisms responsible for shape distortions. Based on the observations, a process model including a new mechanical constitutive model for predictions of residual stresses and shape distortions was proposed and implemented in a general purpose FE-program, as presented in the second paper. In the third paper, the model was validated by comparing spring- in predictions with the experimental results of the first paper complemented by same new experiments. The third paper also embraces a numerical investigation of the effect from the mechanical boundary conditions during cure. So far (in the three first papers), the curing conditions were kept isothermal. When a thick component is cured, the conditions are no longer isothermal owing to heat generated by the exothermal cure reaction. Hence, in the fourth paper the process model was validated against experimental results for a non-isothermally cured component. Finally, in the last paper shape distortions of a complex aircraft component was studied. This was made to both get further validation of the process model as well as investigate the feasibility to simulate large parts of complex shape.

i PREFACE

The work presented in this thesis has been carried out at SICOMP AB in Piteå. A major part of the work has been done in collaboration with Saab AB and is a part of a national aeronautic research program (NFFP) funded by the Defence Materials Administration in Sweden (FMV). Additional support has been provided by the Swedish Foundation for Strategic Research through the Integral Vehicle Structure research school (IVS), the SICOMP foundation and IRECO. I would like to thank my colleagues at SICOMP AB, Division of Polymer Engineering at the Luleå University of Technology and especially my industrial supervisor Dr. Anders Holmberg for their support.

Piteå, October 2002

Magnus Svanberg

ii LIST OF PAPERS

This thesis includes an introduction and the following five papers:

A. Svanberg JM, Holmberg JA. An experimental investigation on mechanisms for manufacturing induced shape distortion in homogenous balanced laminates. Composites Part A: Applied Science and Manufacturing 2001;32(6):827-838.

B. Svanberg JM, Holmberg JA. Prediction of shape distortions, Part I. FE- implementation of a path dependent constitutive model. Submitted Composites Part A: Applied Science and Manufacturing 2002.

C. Svanberg JM, Holmberg JA. Prediction of shape distortions, Part II. Experimental validation and analysis of boundary conditions. Submitted Composites Part A: Applied Science and Manufacturing 2002.

D. Svanberg JM. Shape distortion of non-isothermally cured composite angle bracket. Accepted Plastics, Rubber and Composites 2002.

E. Svanberg JM, Altkvist C, Nyman T. Prediction of shape distortions for a curved composite C-spar. To be submitted.

Parts of paper A, C and D have been presented at following conferences:

• Svanberg M, Holmberg JA. Influence from Cure Schedule on Shape Distortion of RTM Composites. Proc. ICAC 1999, The Sixth International Conference on Automated Composites, Bristol, September 23-24 1999, 145-152.

• Svanberg JM, Holmberg JA. Shape distortion during manufacturing of thermoset composites. SAMPE Europe Students’Conference, Paris, April 15-20, 2000.

• Svanberg JM. Shape distortion of a non-isothermally cured composite angle bracket. Proceedings of FRC 2002, Ninth International Conference on Fibre Reinforced Composites, University of Newcastle upon Tyne, March 26–28 2002, 329-336.

iii CONTENTS

ABSTRACT ...... i PREFACE...... ii LIST OF PAPERS...... iii CONTENTS ...... iv Introduction ...... 1 Factors affecting shape distortions and residuals stresses...... 4 Lay-up...... 4 Anisotropy ...... 6 Material behaviour during cure ...... 7 Expansional effects...... 7 Fibre content...... 8 Cure cycle...... 9 Cure time and temperature ...... 9 Cool-Down rate ...... 10 The effects from the mould and production effects...... 11 Mould surface...... 11 Mould material ...... 11 Radius of mould corners...... 12 Void content ...... 12 Summary of different mechanisms that affect shape distortion ...... 12 Introduction of the papers...... 13

Appended papers Number of pages Paper A 12 Paper B 21 Paper C 27 Paper D 19 Paper E 22

iv Introduction

Polymer composite materials consist of stiff fibres embedded in a polymer matrix and are characterised by a combination of low weight and high stiffness and strength. The properties of the material are governed by the properties of the constituent materials, fibre architecture, volume fraction of fibre etc. The most common fibres are carbon and glass, where carbon fibre is the most exclusive with very high stiffness and low density. In comparison, glass fibre has lower stiffness and higher density but is on the other hand much cheaper. Glass fibres are also an electrical insulation material, and therefore commonly used as structural material in electrical applications. Two types of matrices are used in polymer composites, namely thermoplastics such as Nylon, polypropylene, PEEK etc. and thermosets. Common types of thermosets are polyester, vinylester and epoxy. This work is focused on high performance composites based on epoxy matrix and high fractions of continuous carbon or glass fibres. High performance composites can be manufactured by a number of processes; in common for most of them are that the consolidation of the fibres and matrix is done at the same time as the component is shaped. One of them is RTM (Resin Transfer Moulding) illustrated in Figure 1, where dry reinforcement is placed in a mould cavity. Liquid resin is then injected into the cavity and the reinforcement is impregnated. After the impregnation, the material is kept in the mould for a sufficient time to allow the resin to transform from a liquid to solid material. This event is called the curing and a high performance composite usually requires an elevated temperature to start and maintain the curing reaction. Finally, after complete cure the mould is cooled and opened and the component released, the so-called demoulding. The cure step described so far is frequently called the in-mould cure and is the first step in a cure cycle. Sometimes it is enough with one cure step, the in-mould cure, but usually an additional cure step is needed to completely cure the material. This is referred to as the post cure and is typically performed freestanding in an oven at a higher temperature than was used during the in-mould cure.

1 Tool Resin Air Air Fibres

Preforming Injection Curing Demoulding Figure 1. Illustration of RTM.

The final quality of a composite component is dependent on a number of manufacturing related factors such as voids, dry spots, high temperature peaks, cracking and large shape distortions. For these reasons a good understanding of the materials and manufacturing process is crucial to avoid problems and achieve high quality components. Consequently, the manufacturing processes for thermoset composites are subject to extensive research and during the last decades process models have been developed and verified for mould filling/consolidation and cure for several manufacturing methods. These models are also to some extent available in commercial process simulation tools even though they have not found a widespread industrial use so far. Another related less developed area is modelling residual stresses and shape distortions, which this work focuses on. Residual stresses that develop during manufacturing of thermoset composites have a direct influence on the product quality and can now and then cause trouble. One of the main factors responsible for residual stresses and shape distortions on a high performance composite is thermal shrinkage that occur when the material is cooled from the cure temperature to room temperature. Another is chemical shrinkage of the thermoset matrix, which occurs during the transformation from a liquid to a solid material. During cure, the part is constrained by the mould and the thermal and chemical shrinkage result in residual stresses, which in some occasions can ruin a component as shown in Figure 2. Even if the stresses are not high enough to destroy the component directly, high residual stresses still can cause problems later on, in form of significant reduction of structural strength. At demoulding, cure stresses are released entirely or partially and shape distortions are formed, Figure 3 shows an example of predicted shape distortion of a box.

2 Figure 2. Delaminations caused by residuals stresses.

Figure 3. Predicted shape distortion, quarter model of a box, displacement magnification = 10.

When a component with high shape tolerances is manufactured, the geometry of the mould has to be compensated for the shape distortions. Usually there is no problem to compensate a single curved geometry, such as an angle bracket,

3 based on experience and rules of thumb. Nevertheless, when the component geometry become more complex the compensation of the mould for shape distortions is much harder and sometimes, a number of modification cycles or even complete redesign of the mould is necessary before the right shape of the component is achieved. The possibility of performing this task with assistance of a simulation tool is considered as an important step towards a more effective composite manufacturing. Developing such a simulation tool is the main objective of this thesis, consisting of five papers. Before the papers are introduced, a brief review over factors known from the literature to affect shape distortions and residual stresses are presented.

Factors affecting shape distortions and residuals stresses

The fundamental source for shape distortion is free expansion or contraction of the material. A familiar example is that the length, L of a component made of a homogenous and isotropic material subjected to a temperature change ∆T will change to L⋅(1+α⋅∆T),whereα is the coefficient of thermal expansion. Depending on material, also other factors than temperature may create expansional strains and of particular interest for this thesis is shrinkage due to chemical reactions. Other examples are hygroscopic swelling and physical ageing. The type and amount of shape distortion that result from the expansional strains depend on a number of factors that will be discussed below.

Lay-up

High performances composite laminates usually consists of unidirectional plies with different orientations that are stacked together, to obtain a material with desired properties. Figure 4 illustrates a balanced and symmetric lay-up. A balanced laminate implies that shear strain and shear force are uncoupled from normal strain and force. In a symmetric lay-up forces and midplane strains are uncoupled from moments and bending.

4 0degreeply 90 degree ply 90 degree ply 0degreeply

Figure 4. Illustration of a balanced and symmetric laminate.

A unidirectional ply has much larger thermal expansion in the transverse direction than in the axial direction, due to the difference in thermal expansion between fibre and matrix [1]. The stiffness is also much lower in the transverse direction. Consequently, a temperature change leads to the transverse expansion of each ply being strongly dependent on the orientation of other plies. If the lay-up of a flat plate laminate is balanced but unsymmetrical, a bent or warped shape can be expected at demoulding [1, 2]. This is very clear when a 90° and a 0° ply are cured together, the curvature becomes extremely large due to the large difference in thermal expansion between transverse and axial directions, this situation is illustrated in Figure 5. The stable shape of such a laminate is dependent on the size. A small squared laminate shows a saddle shape and as the size gets larger, the stable shape changes from saddle to a single curved shape [2].

Figure 5. Thermal deformation of an unsymmetrical lay-up. Left and right figure shows undeformed and deformed shape, respectively. In contrast a symmetric lay-up such as [0°/90°/90°/0°], results in a plate without curvature after production, since the in-plane stresses are balanced through the thickness. However, even if shape distortion in form of warpage does not occur, there will always be meso-scale residual stresses associated with the mismatch in ply properties.

5 Anisotropy

Another source of shape distortions is anisotropy or orthotropy in free expansion of the material. As discussed before a plate with a symmetric and balanced lay- up will not show any distortion. A single curved part with symmetric and balanced lay-up on the other hand will distort, for example when it is subjected to a temperature change ∆T. The external angle will than change from θ to θ+∆θ as shown in Figure 6.

∆θ

r θ

Figure 6. Thermally distorted angle section.

Radford and Rennick [3] showed that the spring-in angle, ∆θ for such component is given by,

αα−⋅∆T ∆=⋅θθ()IT +⋅∆α (1) (1T T ) α α where I and T are the coefficients of thermal expansion in the in-plane and through-thickness directions, respectively. ∆θ, θ and ∆T are spring-in angle, angle surrounded by the bend and change in temperature respectively. In the derivation of Eqn. (1) it is assumed that the material properties are orthotropic and uniform through the thickness. The spring-in angle in Eqn. (1) is only dependent on changes in temperature, but also cure shrinkage give a significant contribution. Radford and Rennick have suggested an extension of Eqn. (1) that accounts for the shrinkage during manufacturing,

6 ()αα−⋅∆T () φφ − ∆=θθIT + IT +⋅∆αφ + (2) (1TTT ) (1 ) φ φ where I and T are the manufacturing shrinkage in the in-plane and through- thickness directions, respectively. These two values can be obtained from micromechanics if the manufacturing shrinkage values for the fibre (usually zero) and matrix are known.

Material behaviour during cure

The material behaviour during cure is best explained by considering neat resin cure. Uncured resin is usually viscous at room temperature and consists of linear polymer chains; at this point the degree of cure is defined as zero. During cure, the linear polymer chains are cross-linked and the degree of cure, and the glass transition temperature, Tg increases. When an infinite network is formed the polymer reach gelation and transforms from a liquid in to a rubber like solid material [4]. From this point, the material can sustain stress. During cure at temperatures below the ultimate glass transition temperature (Tg∞) another transition will occur when the Tg of the polymer equals the cure temperature, called the vitrification [4]. Prior this point the polymer is a rubber like solid, which transforms into a solid in the glassy state. This transformation means a dramatic change in mechanical properties. Young’s modulus and the shear modulus in rubbery state are approximately 1% of the modulus in the glassy state. However, not all properties change that dramatically, the coefficient of thermal expansion and bulk modulus are still of the same order of magnitude [5, 6]. Another important aspect of the cure reaction is that a considerable amount of energy is released from the thermoset during curing, which may result in exothermal temperature peaks and gradients in temperature and degree of cure. This is in general not a problem for a thin walled structure, but may be a problem for a thick walled structure and when the mould has poor heat transfer properties.

Expansional effects One of the most important sources of residual stresses and shape distortions is the thermal contraction, which occurs during cooling from cure temperature to room temperature [1]. For most composite systems, the fibre has smaller thermal expansion than the polymer matrix. When analysing the stress in the fibre direction this result is compression in the fibres and tension in the matrix (micro-scale residual stresses). Problems associated with thermal stresses are

7 more severe for laminates, because there is not only a thermal strain between fibre and matrix, but also between the individual plies of the laminate (meso- scale residual stresses). The coefficient of thermal expansion of a thermoset is approximately 2-3 times [5] higher in the rubbery state than in the glassy state. In a 0°/90° laminate, this has no major affect on the in-plane coefficient of thermal expansion but will significantly affect the through-thickness coefficient of thermal expansion and increases the anisotropy of the free expansion of the material. From Eqn. (1) it is clear that for a L-shaped composite the spring-in is increasing faster during cooling in rubbery state than in the glassy state. The effect on stresses from expansional strains developed in rubbery state in a one- or two-dimensionally constrained thermoset is very low due to the low stiffness. Lange et al. [7] for instance did not detect any residual stresses neither from cure shrinkage nor cooling in rubbery state during isothermal curing of epoxy films above their ultimate glass transition temperature. It was only cooling from the glass transition temperature, in the glassy state, that contributed to the residual stresses at room temperature. During volumetrically constrained cure on the other hand the effect from both cure shrinkage and cooling in both the rubbery and the glassy state are important because the bulk modulus is of the same order of magnitude in both glassy and rubbery state [6]. A thermoset shrinks chemically during cure owing to cross-linking of linear polymer chains leading to a denser three-dimensional structure. In the beginning of the reaction, when the polymer is a flowing liquid, cure shrinkage occurs without stress development. At gelation the liquid transforms to a solid and shrinkage results in mictrostresses in a composite owing to the constraint imposed by the fibres. A common approximation is to assume chemical strain linearly dependent on degree of cure [8]. Dependent of the type of thermoset (polyester gels very early and epoxies later) the amount of shrinkage that cause micro stresses differs. The chemical shrinkage of the resin is also an important factor on the macro- scale. For instance Holmberg [9] made experiments with carbon/epoxy U-beams and observed that chemical shrinkage has an important effect on shape distortion and cannot be ignored. Prasatya et al. [10] showed by cure simulations of a three dimensional neat epoxy resin that cure shrinkage contribute up to 30 % of the total residual stresses.

Fibre content A change in fibre volume fraction influences almost all elastic and thermal properties of a unidirectional layer. Wiersma et al. [11] used an elastic model and investigated the effect from changes in volume fraction on shape distortion

8 of a L-shaped part manufactured by prepreg. Sensitivity analysis showed that the change in thermal expansion has greatest effect on the shape distortion. From experiments with carbon/epoxy, Sung and Hilton [12] observed that the fibre content affected both maximum exothermic temperature and degree of cure in thick laminates. Low fibre content implies a high amount of heat per unit mass generated during cure and a high temperature peak in the middle of a carbon/epoxy laminate. This leads to gradients in the expansional strains, which affects the shape distortion. There is also another type of inhomogeneous distribution of fibre and matrix, namely gradients in fibre content. This means an inhomogeneous fibre distribution through the thickness, which is more critical for residual stress and shape distortions, because of the corresponding gradient in mechanical properties through the thickness of the laminate. This phenomenon is common in curved composite sections manufactured with only one stiff mould half, where local corner thinning can appear during manufacturing [13].

Cure cycle

Cure time and temperature In an industrial process, it is often economically advantageous to minimise the cure time by increasing the cure temperature and obtain a faster crosslinking reaction. If this is done without consideration of residual stresses and shape distortions, it is possible that this will cause problems in the form of delaminations or large shape distortions. In general, residual stresses and shape distortions will increase with increased cure temperature [10, 14, 15]. This is also consistent with the results in the first paper in this thesis [16]. More precisely it is actually the temperature at gelation, where the polymer transforms from a liquid to solid, that should be as low as possible, because it is the difference between this temperature and room temperature that produce the thermal part of the residual stresses and/or shape distortions. Dependent on the resin and practical reasons, there are of course limitations of how low cure temperature can be used. For instance a low temperature results in high viscosity, which may effect the impregnation of the fibres during the injection phase in RTM. Another aspect is that the temperature has to be high enough to start and maintain the cross-linking reaction. As previously mentioned residual stresses developed in rubbery state in a one- or two-dimensionally constrained thermoset is very low due to the low stiffness. On the other hand, during volumetrically constrained cure the effect from both cure shrinkage and cooling in both the rubbery and the glassy state are important. Plepys and Farris [6] performed experiments were they cured a neat

9 epoxy resin in a thick-walled glass tube isothermally at 100°C and observed cracks in the resin soon after gelation due to cure shrinkage. To overcome the problem they, instead of using a single step cure schedule, gelled the resin at a low temperature and increased the temperature in a linear ramp up to 100°C. During the heating the resin expands due to the large coefficient of thermal expansion, which results in compressive stresses that compensates for the tension developed by the cure shrinkage. Prasatya et al. [10] has also recently showed this by simulations. This is not only valid for three-dimensionally constrained neat resins it is also relevant for thermoset composites. Consequently, to reduce problems owing to residual stresses and shape distortions, in theory, it is not only advantageous to gel a thermoset composite at as low temperature as possible it is also advantageous to ramp the temperature to the second cure temperature to thermally compensate for the cure shrinkage. Another aspect of the temperature is gradients during cure, which is of special importance when curing thick laminates, where temperature and degree of cure gradients develop across the thickness of the laminate [8, 17]. The free expansion will then vary spatially due to the temperature and degree of cure gradients. Additionally, the variation in degree of cure leads to mechanical properties that develop non-uniformly through the laminate thickness. Chemical shrinkage also occurs non-uniformly because of the spatial variation of the degree of cure. Gradients will lead to a different state of residual stresses and shape distortions compared to a thin laminate where the temperature distribution during cure is approximately uniform.

Cool-Down rate Sarrazin et al. [15] used two different cooling rates (4.2°C/min and 0.6°C/min) during cooling of carbon epoxy prepreg laminates after complete cure and found that the lower cooling rate resulted in less shape distortion. This has also been reported by White and Hahn [14] for carbon/BMI system. They used two different cooling rates, 0.56°C/min respectively 5.6°C/min and found that the lower cooling rate resulted in less shape distortion after in-mould cure but after a following freestanding post cure the difference vanished. They concluded that a slower cooling rate increases the stress relaxation effect. In another paper [18], they found that sufficient time must be allowed for stress relaxation to occur. If a fast cool-down were to occur during or shortly after the chemical shrinkage has completed, a significant portion of residual stress would remain.

10 The effects from the mould and production effects

In Ref. (9) Holmberg discussed a frozen-in deformation when manufacturing carbon/epoxy U-beams due to effect from the mould during in-mould cure. During in-mould cure, the polymer composite is constrained by the mould and shape distortions are formed by release of residual stresses during demoulding. If the composite, on the other hand is free to move during cure, shape distortions may be formed directly without development of macroscopic residual stresses, at least for single curved geometries. The amount of shape distortion after cure in these two cases may differ even for identical cure cycles. The free part is expected to have a higher amount of shape distortions than the constrained. This is because deformations might be frozen into the part when the material transforms from the rubbery to the glassy state (vitrification) during constrained cure. During postcure on the other hand, the specimens are free standing. When the specimen after in-mould cure transforms for the first time from the glassy to rubbery state, the frozen strains are recovered with a corresponding increase in shape distortion [19]. This mechanism has been observed by Kolkman [20] and discussed by Kominar [21].

Mould surface In a resent work by Fernlund et al. [22], spring-in of C-channels manufactured by carbon/epoxy prepregs on an aluminium tool was investigated. They found that the spring-in was 20% less for a tool with no release agent compared to a tool with release agent. Cho and co-workers [2] performed experiments with an aluminium tool with smooth surface, a rough aluminium tool and a rubber tool.

The experiments showed that [0n/90n] carbon/epoxy laminate plates manufactured by the smooth surface tool gave the largest shape distortion and the rubber tool gave smallest shape distortion. They concluded that the slippage effect between the tool and laminate is largest in the rubbery tool respectively lowest in the smooth aluminium tool. These results are however in contrast with the experiments of Sarrazin et al. [15]. During vacuum bagging, they used Teflon to improve the surface finish of the mould. To investigate if the sliding affects shape distortion, they cured laminates with different Teflon arrangements. The result indicates that the Teflon layers close to the composite did not have a marked effect on shape distortion.

Mould material It has been observed that shape distortions may be different for moulds made from different materials. Sarrazin et al. [15] investigated this by curing plates in aluminium and ceramic moulds. They controlled the surface temperature of the plate so it was equal for the two moulds. When this temperature condition was

11 enforced, the shape distortion was the same for the two moulds. They concluded that the apparent effect of the mould materials on shape distortion is most likely due to experiments in moulds with different thermal conductivities, which affects the temperature of the part and hence alters the shape distortions. RTM moulds are usually made of steel or aluminium and there are a mismatch between the thermal expansion of the mould and the composite. Therefore, cooling and heating of the mould affects shape distortion and residual stresses.

Radius of mould corners Holmberg [9] showed using analyses and experiments with U-beams manufactured by a RTM tool that the radii do not effect the spring-in angle of a homogeneous orthotropic material, which is in agreement with Eqn. (1). However, when a beam is produced by autoclave, only one mould half is stiff. This gives local corner thinning during processing [3] and change in thickness means change in fibre volume fraction which affects spring-in. Radford and Rennick [3] observed that the larger corner radii, the less process effects other than cure shrinkage contribute to the shape distortion.

Void content There are two main causes of voids [1]. The first is captured air and secondly voids produced by evaporation during the curing cycle. The second type may be a residual solvent, a product of chemical reactions etc. The void content and distribution depends of fibre volume fraction, resin properties, processing method and processing conditions such as temperature, pressure, time etc. It is well known from the literature that the interlaminar strength of composite laminates decrease significantly with increasing void content [23]. However, also other properties depend to some extent on the void content, e.g. stiffness and thermal expansion, which can be shown by micromechanics. This means that the shape distortion is likely to be affected by variations in void content, which is also indicated by Holmberg [9]. The situation is also complicated because the voids are generally not homogeneous in size and are not evenly distributed in the composite.

Summary of different mechanisms that affect shape distortion

Table 1 contains a summary over factors known from the literature to affect shape distortion and the effect of them.

12 Table 1. Factors known to effect shape distortions and residual stresses. Factors Effect1 Thermal expansion Large Cure shrinkage Large Laminate lay-up Large Cure temperature Large Mould thermal expansion Large Void content Medium Thermal gradients Medium Fibre content gradients Medium Cool-down rate Medium Cure time Small Fibre content Small Mould material surface Small Mould corner radius No2 Mould thermal conductivity No3

1. The effect of the factors is classified in Large, Medium, Small and No effect on shape distortions. 2. May be important if this leads to a local thickness and fibre content variation. 3. The mould thermal conductivity is important if it is low. This may lead to unexpected temperature variations at the mould surface.

Introduction of the papers

Paper A focuses on cure induced mechanisms responsible for shape distortion in homogeneous balanced laminates manufactured with RTM under isothermal conditions. By manufacturing L-shaped angle brackets with the same procedure at different in-mould cure temperatures a majority of the factors known from the literature to affect shape distortions, presented in Table 1, are constant or not present in the experiments. This implies that effects from the cure temperature, thermal expansion and chemical shrinkage can be investigated and separated. The main conclusion of this paper is that constitutive models for modelling of residual stress development and shape distortions of the curing matrix must be able to treat thermal effects, including glass-rubber transitions, chemical shrinkage as well as frozen-in deformations caused by constrained in-mould cure. Finally, explicit rate dependence in terms of creep and relaxation is not important for the manufacturing process and material used here. With starting point in the experimental observations in Paper A a process model for prediction of shape distortions during composite manufacture is presented in Paper B. Particular emphasis is placed on development of a

13 simplified mechanical constitutive model. This paper is the first of two companion papers where the second paper, Paper C in this thesis, is attributed to validation of the process model against experimental results. The mechanical behaviour of a thermoset polymer during cure is viscoelastic and viscoelastic models have been used by many researchers [10, 24, 25, 26, 27, 28, 29]. For instance Kim and White [25] and White and Kim [26] showed that the behaviour of neat resin, respectively laminates of Hercules 3501-6, during cure is viscoelastic and thermorheologically complex. The viscoelastic behaviour during cure and in particular thermorheologically complex behaviour is difficult to characterise and model accurately for a composite. To obtain a simpler description others have approximated the viscoelastic behaviour as thermorheologically simple e.g. [10, 27, 28, 29]. Also this description has drawbacks because use of anisotropic viscoelastic models demands extensive material characterisation and for numerical simulations, they lead to long calculation times and require large memory for storage of internal state variables. For these reasons also elastic models have been used for residual stress calculation. For instance, Bogetti and Gillespie [8], Huang et al. [30] and Johnston et al. [31] used incremental elastic relations. An incremental elastic model, manages to handle a rubber to glass transition, which occurs during the first cure step, in a realistic way. During heating to the second cure temperature a glass to rubber transition might occur, which this model does not manage to handle. In contrary, a linear elastic model manages to handle the glass- rubber but not the rubber-glass transition. Holmberg [9] used an elastic relation and introduced a plastic strain term to handle the rubber to glass transition in a realistic manner. The simplified mechanical constitutive model presented in Paper B related to the model proposed by Holmberg [9] but developed in incremental form for a homogenous anisotropic material and thereby appropriate for a curing resin or curing composites. The model is derived from linear visco-elasticity but rate dependence is replaced by a path dependence on the state variables: strain, degree of cure and temperature. The model is not as general as a viscoelastic model but seems suitable for the material and process conditions typical for RTM because the model captures the mechanisms that were identified in Paper A. The model was also implemented in a general purpose FE-program and verified against analytical test cases. As compared to cure simulations using a fully viscoelastic mechanical model the present model loses a bit of accuracy but leads to significant savings in computational time, memory requirements and costs for material characterisation. As compared to elastic models the simplified constitutive model presented here, captures all mechanisms during a two-step cure schedule, which a general elastic model does not.

14 In Paper C the FE-based process model presented in Paper B is used to produce predictions of spring-in for angle brackets cured at different conditions. The predictions are then compared to the experimental results presented in Paper A to validate the model. Comparisons between predicted and experimental shape distortion shows that the used model and simulation approach capture effects from different cure schedules with good accuracy and after a freestanding post cure the overall agreement is excellent. So far (in Paper A to C) the curing conditions were kept isothermal. When a thick component is cured, the conditions are no longer isothermal owing to heat generated by the exothermal cure reaction. That means that not only effects from the cure temperature, thermal expansion and cure shrinkage is of importance, also other effects presented in Table 1 such as thermal gradients trough the thickness during cure will appear. Hence, to further validate the simulation approach, three thick angle brackets were manufactured under non-isothermal conditions, Paper D. The results from the experiment were then compared with predicted values of temperatures, degree of cure and shape distortion. The results show that the simulation slightly underestimatesthedegreeofcureinthe angle bracket. The temperature in the middle of the laminate agrees well with measured temperatures and the measured values of spring-in agree with the predicted results. Spring-in after the first cure step was predicted using two different boundary conditions, free or fully constrained and it is shown that, the different mechanical boundary conditions result in different spring-in predictions after in-mould cure.

Manufacturing induced deformations (Displacement magnification factor = 10)

Undeformed (tool) shape

15 Figure 7. Global deformations of the air craft component. Finally, in the last paper, shape distortions of a complex aircraft component were studied, see Figure 7. That was possible thanks to BAESYSTEMS on behalf of the PRECIMOULD consortium who made the geometry and the experimental results available. The main objective of the last paper is to validate the simplified constitutive model and the simulation approach presented in Papers B and C for a different material system and geometry, and also to investigate the feasibility for simulation of shape distortions of large components of complex shape that require large FE-models. The results show, that for the material and manufacturing process considered here, the simulations give a good estimate of cure induced shape distortions, even if a number of assumptions and simplifications were used, such as • a simplified constitutive model, • assumed isothermal homogenous curing conditions, • idealised cure schedule, • laminate material properties estimated from the constituent materials, • a simplified interaction between the mould and the component

References

1. Hull D. An introduction to composite materials. Cambridge University Press, 1988. 2. Cho M, Kim MH, Choi HS, Chung CH, Ahn KJ, Eom YS. A Study on the Room-Temperature Curvature Shapes of Unsymmetrical Laminates Including Slippage Effects. Journal of Composite Materials 1998;32(5):460- 483. 3. Radford DW, Rennick TS. Separating Sources of Manufacturing Distortion in Lamianted Composites. Journal of Plastics and Composites 2000;19(8):621- 641 4. Aronhime MT, Gillham JK. Time-Temperature-Transformation (TTT) Cure Diagram of Thermosetting Polymeric Systems. In: Dusek K., editor. Epoxy Resins and Composites III, Berlin: Springer-Verlag, 1986:83-113. 5. Shimbo M, Ochi M, Shigeta Y. Shrinkage and Internal Stresses during Curing of Epoxide Resins. Journal of Applied Polymer Science 1981;26:2265-2277. 6. Plepys AR, Farris RJ. Evolution of residual stresses in three-dimensionally constrained epoxy resin. Polymer 1990;31:1932-1936. 7. Lange J, Toll S, Månson JAE, Hult A. Residual stress build-up in thermoset films cured above their ultimate glass transition temperature. Polymer 1995;36(16):3135-3141.

16 8. Bogetti TA, Gillespie JW. Process-Induced Stress and Deformations in Thick- Section Thermoset Composite Laminates. Journal of Composite Materials 1992;26(5):626-660 9. Holmberg JA. Influence of Chemical Shrinkage on Shape Distortion of RTM Composites. Proc 19th International SAMPE European Conference of the Society for the Advancement of Material and Process Engineering, Paris, France, 22-24 April, 1998. p. 621-632. 10. Prasatya P, McKenna GB, Simon SL. A Viscoelastic Model for Predicting Isotropic Residual Stresses in Thermosetting Materials: Effects of Processing Parameters. Journal of Composite Materials 2001; 35:826-849. 11. Wiersma HW, Peeters LJB, Akkerman R. Prediction of springforward in continous-fiber/polymer L-shaped parts. Composites Part A: Applied Science and Manufacturing 1998;29A:1333-1342. 12. Yi S, Hilton HH. Effects of Thermo-Mechanical Properties of Composites on Viscosity, Temperature and Degree of Cure in Thick Thermosetting Composite Laminate during Curing Processes. Journal of Composite Materials 1998;32(7):600-622 13. Akkerman R, Wiersma HW, Peeters LJB. Spring-forward in continuous fibre reinforced thermosets. Simulation of Materials Processing: Theory, Methods and Applications: Numiform'98, J. Huétink and F.P.T. Baaijens (Eds.), Balkema, Rotterdam, 1998, 471 – 476. 14. White SR, Hahn HT. Cure Cycle Optimization for the Reduction of Processing-Induced Residual Stresses in Composite Materials. Journal of Composite Materials 1993;27(14):1352-1378 15. Sarrazin H, Beomkeum K, Ahn SH, Springer GS. “Effects of the Processing Temperature and Lay-up on Springback. Journal of Composite Materials 1995;29(10):1278-1294 16. Svanberg JM, Holmberg JA. An Experimental Investigation on Mechanisms for Manufacturing Induced Shape Distortions in Homogeneous and Balanced Laminates. Composites Part A: Applied Science and Manufacturing 2001;32(6):827-838. 17. Kim YK, White SR. Viscoelastic analysis of processing-induced residual stresses in thick composite laminates. Mechanics of Composite Materials and Structures 1997;4:361-387 18. White SR, Hahn HT. Process Modelling of Composite Materials: Residual Stress Development during Cure, Part II Experimental Validation. Journal of Composite Materials 1992;26(16):2423-2453 19. Negahban M. Preliminary results on an effort to characterize thermo- mechanical response of amorphous polymers in the glass-transition range.

17 Proc. MD-Vol. 68/AMD-Vol. 215, Mechanics of Plastics and Plastic Composites, ASME, 1995, p.133-152. 20. Kolkman J. Testing epoxide castings by thermal analysis. Holectecniek, 1974;4(2):29-35. 21. Kominar V. Thermo-Mechanical Regulation of Residual Stresses in Polymers and Polymer Composites. Journal of Composite Materials 1996;30(3):406-415. 22. Fernlund G, Rahman N, Courdji R, Bresslauer M, Poursartip A, Willden K, Nelson K. Experimental and numerical study of the effect of cure cycle, tool surface, geometry, and lay-up on the dimensional fidelity of autoclave- processed composite parts. Composites Part A: Applied Science and Manufacturing 2002;33:341-351. 23. Judd NC, Wright WW. Voids and their effects on the mechanical properties of composite-an appraisal. SAMPE journal 1978;Jan/Feb:10- 14 24. White SR, Hahn HT. Process Modeling of Composite Materials: Residual Stress Development during Cure, Part I. Model Formulation. Journal of Composite Materials, 1992;26:2402-2422. 25. Kim YK, White SR. Stress relaxation of 3501-6 Epoxy Resin During Cure. Polymer Engineering and Science 1996;36(23):2852-2862. 26. White SR, Kim KK. Process-Induced Residual Stress Analysis of AS4/3501- 6 Composite Material. Mechanics of Composite Material and Structures 1998;5:153-186. 27. Poon H, Koric S, Ahmad MF. Towards a Complete Three Dimensional Cure Simulation of Thermosetting Composites. International Conference on advanced Composites 1998. 28. Adolf D, Martin JE. Calculation of stress in Crosslinking Polymers. Journal of Composite Materials 1996;30:13-34. 29. Zhu Q, Geubelle PH. Dimensional Accuracy of Thermoset Composites: Shape Optimization. Journal of Composite Materials, 2002;36(6):647-672. 30. Huang X, Gillespie JW, Bogetti T. Process induced stress for woven fabric thick section composite structures. Composite Structures 2000;49:303-312. 31. Johnston A, Vazari R, Poursartip A. A Plane Strain Model for Process- Induced Deformation of Laminated Composite Structures. Journal of Composite Materials 2001;35(16):1435-1469.

18 Paper A

An Experimental Investigation on Mechanisms for Manufacturing Induced Shape Distortions in Homogeneous and Balanced Laminates

Paper A Composites: Part A 32 (2001) 827±838 www.elsevier.com/locate/compositesa

An experimental investigation on mechanisms for manufacturing induced shape distortions in homogeneous and balanced laminates

J.M. Svanberga,b,*, J.A. Holmberga

aSICOMP AB, Swedish Institute of Composites, Box 271, SE-941 26 PiteaÊ, Sweden bDivision of Polymer Engineering, LuleaÊ University of Technology, SE-971 87 LuleaÊ, Sweden Received 6 April 2000; revised 7 November 2000; accepted 10 November 2000

Abstract Manufacturing induced shape distortions is a common problem for composite manufacturers. For single curved geometries the phenom- enon is known as spring-in. Today a lot of effort is spent to develop modelling tools for prediction of spring-in and shape distortions in general. However, good experimental data is rare in the literature and there are no established constitutive models capable of account for the effect from different cure schedules. In this paper, experimental data for spring-in of glass-®bre epoxy composites are presented. The experiments were performed with angle brackets manufactured by RTM, in a steel mould with accurate temperature control. Different in-mould temperature have been used to point out and separate different mechanisms responsible for spring-in. q 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Spring-in; B. Residual/internal stress; E. Cure; A. Laminates

1. Introduction A number of factors affect manufacturing induced shape distortions and researchers have investigated many of them. The manufacturing processes for thermoset composite One of the most important sources of residual stresses and structures are subject to extensive research and during the shape distortions is thermal contraction [6]. The shape last decades process models have been developed and veri- distortion is small when a laminate is cured at low tempera- ®ed for mould ®lling/consolidation and cure for several tures and increases with increasing cure temperature [7,8]. manufacturing methods, [1±4]. These models are also to Another important source of residual stresses and shape some extent available in commercial process simulation distortions is chemical shrinkage of the matrix due to the tools even though they have not found a widespread indus- crosslinking reaction. This means that shape distortion trial use so far. A related area, which is less developed, is increases with increasing degree of cure [8]. Holmberg [9] constitutive modelling of the curing matrix and models for showed during experiments with resin transfer moulded residual stress development and shape distortions. An carbon/epoxy U-beams that for the epoxy in question, exception is the ®lament winding process for which simula- chemical shrinkage has a signi®cant effect on shape distor- tion tools have been developed and are used to predict resi- tion and can not be ignored. Holmberg also discussed a dual stresses [5]. The limited use of process simulation in permanent frozen deformation due to the fact that the U- other composite manufacturing processes and the lack of beam was constrained by the mould during cure. This established methods for prediction of shape distortions mechanism has also been observed by Kolkman [10] and make tool development and manufacturing expensive due discussed by Kominar [11]. to extensive trial and error strategies. Increasing the simula- Other effects known to affect residual stresses and shape tion capacity for these manufacturing processes is an impor- distortion are unsymmetric laminates [12], temperature and tant step towards a more cost ef®cient development and degree of cure gradients [13] and gradients in ®bre content manufacturing of composite structures. [14]. Researchers have also observed that the cooling rate seems to affect the shape distortion, with lower cooling rate resulting in less shape distortion [7]. White and Hahn [8] stated * Corresponding author. SICOMP AB, Swedish Institute of Composites, that a slower cooling rate can be used to allow stress relaxation Box 271, SE-941 26 PiteaÊ, Sweden. Tel.: 146-911-74417; fax: 146-911- 74499. during cooling and results of experiments indicate a reduction E-mail address: [email protected] (J.M. Svanberg). in shape distortion.

1359-835X/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S1359-835X(00)00173-1 828 J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838 experimental results to an approximate model incorporating Nomenclature three mechanisms. DT Change in temperature Spring-in means the increase of the external angle in a u External angle corner of a composite component after curing. Consider the Du Spring-in angle corner of a single curved composite with a through-thick- w Change in spring-in between in-mould cure ness homogeneous orthotropic material. If such a bend is and ®rst postcure subjected to a temperature change DT, the external angle will change from u to u 1 Du, see Fig. 1. Radford and Tg Glass transition temperature Rennick [14] showed that the spring-in angle, Du for such Tcure Cure temperature components can be predicted from geometry and expan- Tin-mould In-mould cure temperature sional strains by Tpost Postcure temperature a Curve ®tting parameter (intersect)  e1 2 e3 b Curve ®tting parameter (slope) Du ˆ u < u e1 2 e3† 1† 1 1 e3 e 1 Free expansion in the plane e 3 Free expansion through the thickness where e 1 and e 3 are the free expansion in the in-plane and e r Matrix strain in rubbery state through-thickness directions, respectively. Du is the spring- e g Matrix strain in glassy state in angle and u is the angle surrounded by the bend. In this e frozen Frozen matrix strain way the expansional strains due to thermal expansion and s vit Matrix stress a vitri®cation cure shrinkage can be accounted for. Before we proceed to Er Young's modulus of the matrix in rubbery experiments we will just make some brief notes about the state thermal and chemical effects that are essential for this Eg Young's modulus of the matrix in glassy state present work. As previously stated, one of the most important sources of residual stresses and shape distortions is the thermal The present paper focuses on cure induced mechanisms contraction, which occurs during cooling from cure responsible for shape distortion in homogeneous balanced temperature to room temperature [8]. laminates manufactured under isothermal conditions with In the calculation of residual stresses it is important to RTM. In Sections 2 and 3 a series of experiments are notice that the coef®cient of thermal expansion is approxi- presented. Effects from thermal expansion, chemical shrink- mately 2±3 times higher in the rubbery than in the glassy age and other potential mechanisms are investigated and state [9,15] and that the Young's modulus for the matrix is separated by measuring spring-in of a number of angle approximately 100 times lower in the rubbery than in the brackets manufactured with the same procedure at different glassy state. The bulk modulus on the other hand has the in-mould cure temperatures. In Section 4, the experimental same order of magnitude in both the glassy and the rubbery results are explained in a qualitative manner by ®tting the state. This means that very low stresses develop in the rubbery state in the case of a uniaxial stress state. In a three-dimensional stress state, however, signi®cant stresses may also develop in the rubbery state [16]. For thermoset polymer composites, the matrix undergoes chemical shrinkage during cure. This has a major effect on the residual stresses within a composite material. Much of the resin shrinkage occurs while the resin is a liquid and therefore without stress development. The proportion of shrinkage in the liquid state depends on the degree of cure at gelation, polyesters gel very early and epoxies have a higher degree of cure. Resin shrinkage in the later stages of cure (after gelation) leads to residual stresses that will cause shape distortions when the part is released from the mould. A common approximation is to assume that the chemical volumetric shrinkage of the resin is linearly dependent on the degree of cure [17].

2. Manufacturing of angle brackets

Angle sections have been manufactured by RTM at a high Fig. 1. Distorted angle section. level of thermal and dimensional control. Different in-mould J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838 829 channels positioned 40mm from the inner mould surface. This design allows for accurate temperature control during the entire cure cycle. The injection strategy for the angle brackets was edge injection perpendicular to the length of the angle bracket. To prevent ªrace trackingº a string of tacky tape was tucked against both ends of the reinforcement before closing the mould. An inlet and outlet hose connects into the upper side of the male half and a channel distributes resin. This injec- tion strategy was chosen to provide a homogeneous quality along the radius of the angle. The inner radius is relatively large, 10mm, to avoid wrinkling and other performing problems in this region. During preforming, 16 layers of the reinforcement were cut and stacked together in the 4 mm thick mould cavity in the female mould half. Degas- Fig. 2. Female mould half. sing of the resin and hardener was performed separately under 70% vacuum in a glass dessicator placed in an oven at 408C for 60min. Then the resin and hardener were mixed temperatures have been used to separate the different thoroughly and degassed in the pressure pot at 408C for an mechanisms responsible for spring-in. additional 10min. Prior to injection, the tool was heated to the in-mould cure temperature. At the resin inlet into the 2.1. Material mould, there was a 1 mm thick and 15 mm wide distribution channel, covering the width of mould cavity. The purpose of The resin used in this study was the cold curing epoxy the channel was to provide a rapid tempering of the resin. laminate system wAraldite LY5052/Hardener HY5052, The injection pressure was 5 bar, which was kept until the which is suitable for resin transfer moulding at a wide resin reached gelation. Angle bracket specimens were in- range of in-mould cure temperatures. The mix ratio used mould cured at 40, 60, 80, respectively, 1008C. The in- was 100 to 38 parts by weight, recommended by Ciba mould curing times were selected suf®ciently long for the [18]. An 8 harness satin glass weave, Hexcel 7781-127, specimens to approach the maximum degree of cure at each was chosen as reinforcement. The surface weight is in-mould temperature used, see Table 1. After complete in- 300 g/m2 with 53% of the ®bres in the warp direction [19]. mould cure the specimens were cooled in the mould from the in-mould cure temperature to room temperature in 2.2. Manufacturing procedure approximately 45 min by ¯ushing cold water through the channels in the mould (which is considered as fast cooling in The injection equipment used to manufacture angle this study). Three specimens in-mould cured at 808C, were bracket specimens was a heated pressure pot positioned on cooled to room temperature without ¯ushing the mould chan- a scale [20]. The RTM-mould used in this study was made nels with cold water (slow cooling). Cooling in this manner of steel and consists of two rigid mould halves, the female took approximately 12 h. After demoulding, the specimens mould half is shown in Fig. 2. The mould was designed were cut to dimensions L50 £ 50mm (Fig. 3). to be very stiff so as to enable manufacture of angle All specimens were postcured free standing in an oven in brackets at a high level of dimensional control. A water 1208C for 4 h and then cooled free standing at room heater connected to channels in the mould was used for temperature, the same postcure schedule was used for all temperature control. Each mould half was seven water specimens. To verify that the specimens are fully cured and that all mechanisms responsible for spring-in have been captured, all specimens were subjected to a second post- Table 1 In-mould cure schedules cure (or annealing) at 1508C for 2 h and cooled free-stand- ing to room temperature. As de®ned by Aronhime and Gill- Num. of Spec.a Cure schedule Cooling T (8C)b g ham [21], 1508C is well above Tg1, for this material. After each cure step, the specimens were placed in a bucket with 2408C/24 h Fast 68 2608C/20h Fast 94 dry silica gel and stored in a freezer, to prevent moisture 3808C/15 h Fast 106 uptake and prevent additional cure, before the spring-in 3808C/15 h Slow 106 angle was measured. 21008C/12 h Fast 116

a Number of specimens manufactured. 2.3. Characterisation b Measured by DSC. Calculated by half Cp extrapolation of the DSC curves. In this study, 12 angle brackets were manufactured in 830 J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838

Fig. 3. Angle bracket. total and the spring-in angle and glass-transition tempera- in Fig. 4 are for specimens cooled with fast cooling rate. The ture were measured after in-mould cure. After postcure and results show that the spring-in increase with increased in- a second postcure, the spring-in was measured. The glass- mould cure temperature and that the spring-in can either transition temperature was measured with a Perkin±Elmer decrease or increase during postcure. DSC7 differential scanning calorimeter. The spring-in angle was measured with a Zeiss Umess measuring machine 3.1. Fast cooled specimens (tolerance 1 mm). The measuring machine was in a tempera- ture and humidity controlled room (208C, 43% RH). The For both angle brackets manufactured with the 408C in- angle was measured at 3 points and the results averaged for mould cure cycle, the spring-in decreases during postcure, each specimen. The measurements were taken 65 mm from see Fig. 4. The specimens in-mould cured at 608C do not each edge and at the centre. Before measurement, the bucket change spring-in during postcure. Three angle brackets have with specimens was kept in the measurement room for more been manufactured with the 808C in-mould cure cycle, for than 2 h to allow the specimens to stabilise at the measure- these specimens postcuring gave an increased spring-in. ment temperature. Finally, two angle brackets were manufactured with the 1008C in-mould cure schedule. Again the spring-in appears to increase during the postcure. 3. Experimental results 3.2. In¯uence from the cooling rate after in-mould cure Tables 2 and 3 show measured spring-in after in-mould cure and postcure for fast cooled and slowly cooled speci- Six angle brackets were manufactured at an in-mould mens, respectively. The second post-cure did not change the cure temperature of 808C. After completed in-mould cure, spring-in as a result was not examined further. In Fig. 4 three of the specimens were rapidly cooled to room average spring-in after in-mould and postcure have been temperature and the other three were slowly cooled to plotted versus in-mould cure temperature. The data shown room temperature. The signi®cance test shown in Tables 4

Table 2 Measured spring-in for specimens in-mould cured at different temperatures, cooled fast to room temperature

Specimen, I In-mould cure schedule Measured spring-in (8)

After in-mould cure, Du 1 After postcure, Du 2

1408C/24 h 0.53 0.30 9408C/24 h 0.64 0.33 2608C/20 h 0.74 0.74 8608C/20 h 0.81 0.81 3808C/15 h 0.99 1.16 6808C/15 h 1.00 1.19 7808C/15 h 1.02 1.26 41008C/12 h 1.74 2.12 10100 8C/12 h 1.35 1.74 J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838 831

Table 3 Measured spring-in for specimens in-mould cured at 808C and cooled slowly to room temperature

Specimen, I In-mould cure schedule Measured spring-in Du (8)

After in-mould cure, Du 1 After postcure, Du 2

5808C/15 h 1.25 1.41 11 808C/15 h 0.98 1.18 12 808C/15 h 0.99 1.17 and 5 shows that a null hypothesis of no difference between spring-in angle. For specimens in-mould cured at 408C the cooling rates is evident (signi®cance levels between 14 and specimens in-mould cured at 608C on the other hand, and 65%). There is no difference in spring-in neither after the spring-in has decreased, respectively, stayed unchanged in-mould cure nor after postcure between rapidly and slowly during postcure. cooled specimens. The reason why specimens in-mould cured at 40and 608C, respectively, behave differently during post-cure can 3.3. Summary of result from spring-in experiments be explained by the fact that the coef®cient of thermal expansion for the epoxy is much higher in the rubbery The change in spring-in for all fast cooled specimens than in the glassy state. As explained before and shown is tabulated in Table 6. An interesting plot of the with Eq. (1), spring-in is a result of different free expansion change in spring-in during postcure versus in-mould cure in the in-plane direction than in the through-thickness direc- temperature is given in Fig. 5. The spring-in decreases after tion. In the in-plane direction the ®bres govern the free postcure for the specimens in-mould cured at 408C, stays expansion, but in the through-thickness direction the matrix unchanged for 608C in-mould cure and increases for dominates the free expansion. This means that the thermal the specimens in-mould cured at 80and 100 8C. These effect of the spring-in angle is stronger in the rubbery state are signi®cant effects and have a con®dence level over than in the glassy state. The angle brackets cured at 40and 95%. 608C, respectively, were in-mould cured at the lowest temperatures, which means small thermal strains and a 3.4. Variation of the thickness low degree of cure after in-mould cure. When the angle brackets are heated up to the postcure temperature the poly- The thickness of the 12 angle brackets manufactured in mer matrix expands in the glassy state until the specimens this study was measured with a micrometer at 6 different glass transition temperature (Tg) is reached and then the points, 4 points on the ¯at part and 2 points at the radius. The expansion continues in the rubbery state. A low Tg (result average thickness is 3.97 mm (standard deviation 0.05 mm) of low degree of cure) means more expansion in the rubbery at the radius and 3.80mm (standard deviation 0.05) at the state during heating to the postcure temperature. After post- ¯at parts. The thickness variations are caused by the design curing at 1208C for 4 h, Tg has passed the postcure tempera- of the mould. If the mould is not completely closed, the ture and the matrix shrinks in a glassy state during cooling to thickness will be larger in the radius than at the ¯at parts. room temperature. This means that the spring-in follows a The ®bre volume fraction was measured to be 49% (using different path during heating to the postcure temperature pyrolysis at 5608C for 4 h) at the ¯at section. This means than during cooling after completed postcure, which in that the ®bre content is approximately 47% by volume at the turn decreases the shape distortion. If this path dependence radius. is stronger than other mechanisms responsible for changes in shape distortion during postcure, such as for instance 3.5. Glass transition temperature further chemical shrinkage which increase the shape distor- tion, this explains decreased or constant spring-in during The glass transition temperature was measured by differ- postcure for the specimens in-mould cured at 40and ential scanning calorimetry (DSC) after in-mould cure for 608C, respectively. In Fig. 6 is a schematic representation each specimen, see Table 1. of the effect of the path dependence and the effect of further chemical shrinkage illustrated during postcure. Line 1 in 4. Discussion Fig. 6 corresponds to decrease of spring-in due to thermal expansion in the glassy state and line 2 corresponds to The difference in spring-in between the 80and 100 8C decrease of spring-in due to thermal expansion in the cure cycles in Fig. 4 agrees with what is expected. The rubbery state. Further chemical shrinkage during postcure spring-in increases with increasing cure temperature since is illustrated by line 3 in Fig. 6. Finally, line 4 represent a high cure temperature results in larger thermal strains and increase of spring-in due to cooling in glassy state from the higher degree of cure. The postcure also increases the postcure temperature to room temperature. 832 J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838 postcure on the other hand, the specimens are free standing. When the specimen after in-mould cure transforms for the ®rst time from the glassy to rubbery state, the frozen strains are totally or partly released with a corresponding increase in spring-in. This mechanism is essentially a recovery of large plastic strains [22]. If the locked in strains are higher at a higher in-mould cure temperature this may explain the higher increase of the spring-in during postcure for the 1008C than the 808C in-mould cured specimens. A model which predicts the behaviour of angle brackets to thermal and cure shrinkage was developed to under- stand the mechanisms involved in this process. The angle Fig. 4. Average spring-in after in-mould cure and postcure for specimens bracket is considered as fully constrained during in-mould cooled fast after in-mould cure. cure and free to move during the postcure. The discussion focuses on the effects from thermal expansion in the glassy A higher in-mould cure temperature means a higher and the rubbery state, chemical shrinkage and frozen degree of cure and a corresponding higher Tg after in- deformations. mould cure, which means small effects of both thermal In a previous work by Holmberg [9] an analytic spring-in expansion in the rubbery state and the effect of further model was developed based on micro-mechanics and three- chemical shrinkage during postcure. If the mechanisms dimensional laminate theory, which accounts for all described above were the only mechanisms responsible mechanisms described above. A micro-mechanical model, for shape distortion, the increase of spring-in during post- similar to the one presented in Ref. [9], should be feasible to cure would be smaller for a specimen in-mould cured at explain the experimental results, but to keep focus on the 1008C than for a specimen in-mould cured at 808C. mechanisms, a simpler approach will be used here instead. However, the experiments show a larger increase of the The purpose of the following discussion is to use a simple spring-in during postcure for specimens in-mould cured at model as a tool to interpret the experimental results in terms 1008C than for specimens in-mould cured at 808C. This of how different mechanisms in¯uence shape distortion. indicates the presence of an additional signi®cant mechan- First, we assume that spring-in is governed by the three ism. A possible mechanism, which is a result of the angle mechanisms suggested above: (1) thermal expansion in brackets being fully constrained during in-mould cure, is the glassy and the rubbery state; (2) chemical shrinkage; frozen-in permanent deformations. These deformations are and (3) frozen-in deformations. We then write locked into the specimen when the specimen transforms from a rubbery to glassy state during in-mould cure. During Du ˆ Du T 1 Du C 1 Du F 2†

Table 4 Signi®cance test on the in¯uence of cooling rate on spring-in for specimens in-mould cured at 808C

Specimen Cooling Spring-in Squared residuals

2 2 2 2 i Du 1 Du 2 d1 ˆ Du1i 2 Du1† d2 ˆ Du2i 2 Du2†

3 Fast 0.998 1.168 3.04 £ 10204 1.83 £ 10203 6 Fast 1.008 1.198 5.24 £ 10206 1.57 £ 10204 7 Fast 1.028 1.268 3.89 £ 10204 3.06 £ 10203 Average, Du Fast 1.008 1.218 5 Slow 1.258 1.418 3.13 £ 10202 2.57 £ 10202 11 Slow 0.988 1.188 8.80 £ 10203 5.24 £ 10203 12 Slow 0.998 1.178 6.92 £ 10203 7.74 £ 10203 Average, Du Slow 1.078 1.258

Sum of squares, Sd2 0.05 0.04 p Std. error, s ˆ Sd 2= 6 2 2†† 0.118 0.108 Dufast 2 Duslow 20.07820.048

0.78 0.49

Du 2 Du Test statistic t ˆ fastrslow 0 1 1 s 1 3 3

Signi®cance level, Pr(utu . t0) 48% 65% J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838 833

Table 5 Table 6 Signi®cance test on the in¯uence of cooling rate on the change in spring-in Change in spring-in during postcure for angle brackets cooled fast after in- during postcure for specimens in-mould cured at 808C mould cure

Specimen i Cooling Change in Squared Specimen i In-mould Change in Squared spring-in residuals cure spring-in w ˆ residuals 2 2 w ˆ d ˆ schedule Du2 2 Du1 (8) d ˆ 2 2 Du2 2 Du1 wi 2 w† wi 2 w†

3 Fast 0.188 6.43 £ 10204 1408C/24 h 20.23 1.30 £ 10203 6 Fast 0.198 1.04 £ 10204 9408C/24 h 20.30 1.30 £ 10203 7 Fast 0.248 1.27 £ 10203 Average 408C/24 h 20.27 Average Fast 0.208 2608C/20 h 0.01 1.36 £ 10205 5 Slow 0.168 2.79 £ 10204 8608C/20 h 0.00 1.36 £ 10205 11 Slow 0.208 4.61 £ 10204 Average 608C/20 h 0.00 12 Slow 0.188 2.27 £ 10205 3808C/15 h 0.18 6.43 £ 10204 Average Slow 0.188 6808C/15 h 0.19 1.04 £ 10204 7808C/15 h 0.24 1.27 £ 10203 Sum of squares, Sd2 0.003 p Average 808C/15 h 0.20 ± Std. error, s ˆ Sd 2= 6 2 2†† 0.038 41008C/12 h 0.38 1.07 £ 10205 10100 8C/12 h 0.38 1.07 £ 10205 w fast 2 w slow 0.028

1.84 Average 1008C/12 h 0.38 ±

w 2 w Test statistic t ˆ fastrslow 2 203 0 Sum of squares, Sd 4.67 £ 10 1 1 s 1 Std. error, s ˆ 0.03 3 3 p Sd 2= 6 2 2††

Signi®cance level, Pr(utu . t0) 14% cure, spring-in will be related to cure temperature by

T T Duin-mould ˆ bg Tin-mould 2 TRT† 4a† where Du T, Du C and Du F are spring-in due to thermal, chemical, respectively, frozen-in strains. The glass transi- Du C ˆ aC 1 bCT T † 4b† tion temperature after ®nal cure is assumed to be a linear in-mould g in-mould function of the cure temperature, which is reasonable if the F F F curing times are long enough to give approximately the Duin-mould ˆ a 1 b Tin-mould 4c† maximum degree of cure at each temperature. The ultimate After postcure, the spring-in components can be described as: glass transition temperature for the resin used in this study is higher than the postcure temperature, which means that this T T T T Dupost ˆ Duin-mould 1 bg 2 br † Tpost 2 Tg Tin-mould†† 5a† crude assumption of Tg is reasonable also for postcure. This gives C C C Dupost ˆ a 1 b ´Tg Tpost† 5b†

Tg Tcure†ˆaT 1 bT Tcure 3† F g g Dupost ˆ 0 5c† T T Duin-mould and Dupost; are thermal spring-in at room tempera- where Tg is the glass transition temperature after isothermal T ture after in-mould cure and postcure, respectively, and bg curing for a long time at the temperature Tcure and aTg and bTg are ®tting parameters. We make the following approximations on the laminate level:

² Neither the glassy nor the rubbery coef®cients of thermal expansion depend on the degree of cure. ² The chemical shrinkage strains are linear functions of cure temperature. ² Neither the glassy nor the rubbery stiffness depend on temperature, time or degree of cure.

By making these approximations the model proposed in Ref. [9] results in expressions where the components of Fig. 5. Change in spring-in during postcure versus in-mould cure tempera- Eq. (2) are linear functions of temperature. After in-mould tures, 95% con®dence limits calculated from the standard error in Table 6. 834 J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838

Table 7 Fitting parameters

Parameter Description Value

a T intersection 41.8 Tg g b T slope 0.8 Tg g T bg Thermal slope in glassy state 0.012 T br Thermal slope in rubbery state 0.030 aC Chemical intersection 20.083 bC Chemical slope 0.007 aF Frozen-intersection 20.097 bF Frozen slope 20.002

Fig. 6. Spring-in during postcure. chemical reaction has stopped. Point C is the resulting T spring-in at room temperature. In reality, the specimen is and br are thermal slope ®tting parameters in the glassy and C C constrained by the mould during the entire in-mould cure rubbery state, respectively. Duin-mould and Dupost are spring-in due to chemical shrinkage after in-mould cure and after post- schedule and the spring-in is formed at demoulding, which cure, aC and bC are intersect and slope ®tting parameters, corresponds to the solid line in Fig. 8. Note that the spring-in respectively, for spring-in caused by chemical shrinkage. angle after in-mould cure in Fig. 8 is not the same for a F F constrained specimen as for a specimen free to move. Duin-mould and Dupost are spring-in due to frozen-in deforma- tions after in-mould cure and after postcure, aF and bF are This summarises the effects of the mechanisms responsible intersect and slope ®tting parameters, respectively, for for spring-in during in-mould cure. The mechanism that causes the frozen deformation can be spring-in due to frozen-in strains. Finally, Tin-mould, Tpost and illustrated by considering uniaxial loading and unloading of TRT are in-mould cure temperature, postcure temperature and room temperature, respectively. a fully cured rod of neat resin, see Fig. 9. Consistent with the The calculated spring-in from the equation system above approximations made in the previous discussion we neglect can then be ®tted to experimental data by minimising the the time dependence that will be present in reality and least square error, which results in ®tting parameters accord- consider the resin to exhibit its full unrelaxed glassy modu- ing to Table 7. Fig. 7 shows experimental spring-in lus in the glassy state and the relaxed rubbery modulus in the compared with calculated spring-in after in-mould cure rubbery state. The line from point 1 to point 2 in Fig. 9 and after postcure, respectively. By using this simple corresponds to loading at a temperature slightly above Tg, model it is then possible to illustrate the mechanisms for in the rubbery state where the stiffness is low. Then the rod shape distortion during manufacturing. is cooled to just below Tg. The second line, from point 2 to point 3, is unloading in the glassy state where the stiffness is 4.1. 408C in-mould cure and 1208C postcure high. The residual strain is frozen into the polymer due to fact that the polymer is not in thermodynamic equilibrium in Fig. 8 presents calculated shape distortions during in- glassy state. The magnitude of the frozen-in strain can be mould cure for an angle bracket manufactured at 408C. If determined from Eq. (6), which can be derived from Fig. 9 the specimen is considered free to move during the in-mould "# cure, the spring-in would follow the dashed line in Fig. 8. 1 1 er ˆ eg 1 efrozen ) efrozen ˆ 2 svit 6† Residual stresses and shape distortions start to develop after Er Eg gelation of the resin, which corresponds to point A. At point

B the material begins to cool to room temperature and the where e r, e g and e frozen are strains in the rubbery, glassy and

Fig. 7. Experimental spring-in compared with predicted spring-in after in-mould cure, respectively, postcure. J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838 835 frozen-in states, respectively. s vit is the stress when the polymer transforms from rubbery to glassy state (vitri®ca- tion). Finally Er and Eg are Young's modulus in the rubbery and glassy states, respectively.

If the rod is then heated without load above its Tg the frozen strain is relaxed, path 3-1 in Fig. 9, which means that thermodynamic equilibrium, is reached. The physical motivation for the presence of this mechanism is that in the rubbery state large molecular segments are rearranged if external loading deforms the polymer. If the polymer then is cooled to the glassy state, where the molecular mobility is limited, these large segments can not return to their equilibrium position when the external load is Fig. 8. Predicted spring-in during 408C in-mould cure. removed. A permanent deformation will be the result in the glassy state. To recover this deformation the polymer has to be heated without external load to the rubbery are 0.50 and 0.278 after in-mould and postcure, respectively state, where the molecular segments can return to their (point C and G in Fig. 10). equilibrium position. This mechanism has been discussed by Kominar [11] and observed by Kolkman [10] in ther- mal analysis of casting epoxide. Kolkman observed a 4.2. 608C in-mould cure and 1208C postcure small length change of test bars cut out from cast epoxide when passing through the glass transition for the ®rst time. Fig. 11 presents calculated spring-in during postcure for

This change of length was much larger if the chemical the 608C in-mould cured specimens. Tg is higher after in- shrinkage of the polymer was constrained by cast-in metal mould cure at 608C than 408C, which means a smaller effect parts. from thermal expansion in rubbery state. The spring-in The postcure is described between point C and G in Fig. calculated by the model are 0.82 and 0.808 after in-mould 10. At point D the matrix transforms from the glassy to and postcure, respectively. rubbery state. The step at point D in Fig. 10is a consequence of the release of frozen deformation during the constrained in-mould cure since the thermoset polymer matrix is not in 4.3. 808C in-mould cure and 1208C postcure thermodynamic equilibrium in the glassy state whereas it rapidly reaches thermodynamic equilibrium in the rubbery During postcure for a specimen in-mould cured at 808C, state. From Fig. 10it is clear that the spring-in angle as a the effect of thermal expansion in rubbery state (D±E in Fig. function of temperature followed a different path during 12) is small compared to the 40and 60 8C in-mould cured heating to the postcure temperature than during cool-down specimens. In this case the spring-in angle increases during after postcure, which gave a decreased spring-in after post- postcure and the model calculated values are 1.13 and 1.328 cure. The spring-in calculated by the model, Eqs. (2)±(5), after in-mould cure and postcure, respectively, see Fig. 12.

Fig. 9. Stress±strain curve, uniaxial loading in rubbery state and unloading in the glassy state. 836 J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838

Fig. 10. Calculated spring-in during postcure for a specimen in-mould cured at 408C.

Fig. 11. Calculated spring-in during postcure for a specimen in-mould cured at 608C.

Fig. 12. Calculated spring-in during postcure for a specimen in-mould cured at 808C.

Fig. 13. Calculated spring-in during postcure for a specimen in-mould cured at 1008C. J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838 837 4.4. 1008C in-mould cure and 1208C postcure Acknowledgements

Finally, when a specimen is in-mould cured at 1008C, Tg The authors gratefully acknowledge the efforts of Runar after in-mould cure is close to the postcure temperature. The LaÊngstroÈm at SICOMP AB for manufacturing angle brack- effect from thermal expansion in the rubbery state has now ets and Anders OÈ hman at ABB Plast AB for measuring the almost disappeared but there is still an effect from the spring-in angles. This work was performed in collaboration release of the frozen strains, see Fig. 13. Calculated with Saab AB and is a part of a national aeronautic research spring-in are 1.45 and 1.858 after in-mould cure and post- program (NFFP) funded by the Defence Materials Admin- cure, respectively. istration in Sweden (FMV).

4.5. The effect of frozen-in strain from the in-mould References temperature

We have previously suggested a mechanism responsible [1] Gebart BR. Analysis of heat transfer and ¯uid ¯ow in the resin trans- fer moulding process. Doctoral thesis 1992:112D. LuleaÊ University of for the step at point D in Figs. 10±13, which was due to Technology, 1993. release of frozen strains during the ®rst transition from [2] LundstroÈm TS, Gebart BR, Lundemo CY. Void formation in RTM. J glassy to rubbery state. In this section we are going to Reinforced Plastics Compos 1993;12:1340±9. discuss the effect from the in-mould temperature on the [3] Kenny JM, Maffezoli A, Nicolais L, Mozzola M. A model for the level of the spring-in step due to this mechanism. The exam- thermal and chemorheological behaviour of thermoset processing: (II) unsaturated polyester based composites. Compos Sci Technol ple with a uniaxially loaded bar (Eq. (6)) shows that the 1990;38:339±58. frozen strain is proportional to the stress at vitri®cation. A [4] Kenny JM. Integration of processing models with control and optimi- high in-mould temperature corresponds to a high degree of zation of polymer composite fabrication. Comput Mech. 1992:529±44. cure and more chemical shrinkage at vitri®cation. This [5] Olofsson K. Stress development in wet ®lament wound pipes. means that the stress level and corresponding frozen strain J Reinforced Plastics Compos 1997;16(4):372±90. [6] Hull D. An introduction to composite materials. Cambridge: at vitri®cation is higher at higher in-mould cure tempera- Cambridge University Press, 1988. tures. This is also the observed result of the analysis of the [7] Sarrazin H, Beomkeum K, Ahn S-H, Springer GS. Effects of the experiments performed in this study. processing temperature and lay-up on springback. J Compos Mater In summary, the analysis of the experimental results 1995;29(10):1278±94. suggests that an accurate model for prediction of shape distor- [8] White SR, Hahn HT. Cure cycle optimization for the reduction of processing-induced residual stresses in composite materials. J tions should include at least three mechanisms: thermal expan- Compos Mater 1993;27(14):1352±78. sion (different in glassy and rubbery state), chemical shrinkage [9] Holmberg JA. In¯uence of chemical shrinkage on shape distortion of and frozen-in deformations. Since there was no difference in RTM composites. Proceedings of the 19th International SAMPE spring-in between rapid and slowly cooled specimens, it European Conference of the Society for the Advancement of Material appears as if it is not necessary to include any explicit time and Process Engineering. Paris, France, 22±24 April 1998. p. 621±32. [10] Kolkman J. Testing epoxide castings by thermal analysis. Holectec- dependence interms of relaxation into the model,as long as the niek 1974;4(2):29±35. cooling rates are moderate. [11] Kominar V. Thermo-mechanical regulation of residual stresses in poly- mers and polymer composites. J Compo Mater 1996;30(3):406±15. [12] Cho M, Kim MH, et al. A study on the room-temperature curvature shapes of unsymmetrical laminates including slippage effects. J 5. Conclusions Compo Mater 1998;32(5):460±83. [13] Kim KK, White SS. Viscoelastic analysis of processing-induced resi- This work has demonstrated how the cure schedule affects dual stresses in thick composite laminates. Mech Compos Mater spring-in. Three mechanisms have been suggested as respon- Struct 1997;4:361±87. sible for the shape distortions: thermal expansion (different in [14] Radford DW, Rennick TS. Determination of manufacturing distortion glassy and rubbery state), chemical shrinkage and frozen-in in laminated composite components. Proceedings of ICCM-11. Gold Coast, Australia, Pt. IV, 14±18 July 1997. p. 302±12. deformations. A simple spring-in model that accounts for [15] Shimbo M, Ochi M, Shigeta Y. Shrinkage and internal stresses during these mechanisms have been used to illustrate and separate curing of epoxide resins. J Appl Polym Sci 1981;26:2265±77. the contribution from each mechanism at different cure [16] Plepys AR, Farris RJ. Evolution of residual stresses in three-dimen- schedules, showing that they are all signi®cant. sionally constrained epoxy resin. Polymer 1990;31:1932±6. The main conclusion of this work is that constitutive [17] Bogetti TA, Gillespie JW. Process-induced stress and deformations in thick-section thermoset composite laminates. J Compos Mater models for the curing matrix and models for residual stress 1992;26(5):626±60. development and shape distortions must be able to treat both [18] Data sheet: araldite LY5052/hardener HY5052. Ciba Polymers, thermal effects, including glass-rubber transitions, and Switzerland, 1994. chemical shrinkage as well as frozen-in deformations [19] Data sheet: reinforcements for composites. Hexcel Fabrics, France, caused by constrained in-mould cure. Explicit time depen- 1997. [20] Sandlund E. Development of injection equipment. SICOMP Techni- dence in terms of creep and relaxation is not as critical for cal Report 96-005. SICOMP AB, Box 271, SE-941 26 PiteaÊ, Sweden, the cooling rates studied in the present paper. 1996. 838 J.M. Svanberg, J.A. Holmberg / Composites: Part A 32 (2001) 827±838

[21] Aronhime MT, Gillham JK. Time±temperature±transformation [22] Negahban M. Preliminary results on an effort to characterize thermo- (TTT) cure diagram of thermosetting polymeric systems. In: Dusek mechanical response of amorphous polymers in the glass-transition K, editor. Epoxy resins and composites, vol. III. Berlin: Springer, range. Proc. MD±Vol. 68/AMD±Vol. 215, Mechanics of Plastics and 1986. p. 83±113 Plastic Composites, ASME, 1995, p. 133±52. Paper B

Prediction of shape distortions

Part I. FE-implementation of a path dependent constitutive model

Paper B Prediction of shape distortions

Part I. FE-implementation of a path dependent constitutive model

J. Magnus Svanberg a, b and J. Anders Holmberg a

aSICOMP AB, Swedish Institute of Composites, Box 271, 941 26 Piteå, Sweden bLuleå University of Technology, Division of Polymer Engineering,971 87 Luleå, Sweden

Abstract

There is a great interest, especially from the aircraft industry, to increase the ability to understand and predict development of shape distortions and residual stresses during manufacture of polymer composite components. This will result in more cost efficient development, improved performance and optimised manufacturing of composites. In a previous work by the authors, it was demonstrated that models for residual stress development and shape distortions must account for thermal expansion (different in glassy and rubbery state), chemical shrinkage due to the crosslinking reaction and finally frozen-in deformations. The present paper presents a simple mechanical constitutive model that accounts for the mechanisms mentioned above. The model is a limiting case of linear viscoelasticity that permits us to replace the rate dependence by a path dependence on the state variables: strain, degree of cure and temperature. This means significant savings in computational time, memory requirements and costs for material characterisation as compared to conventional viscoelastic models. This is the first of two papers, the second paper deals with experimental validation and analysis of mechanical boundary conditions during prediction of shape distortion.

Keywords: B. Residual stress; B. Cure behaviour; A. Finite element analysis (FEA)

1 Introduction

Residual stresses develop during manufacture of polymer composites and have a direct influence on the product quality. For thermoset composites crosslinking of the polymer leads to chemical shrinkage and to this thermal shrinkage is added during cooling from the cure temperature to room temperature. During cure, the part is constrained by the mould and thermal and chemical shrinkage result in residual stresses on all scales, from the micro scale on fibre level to the macro scale on component level. At demoulding, after in-mould cure, macro scale residual stresses are fully or partly released and shape distortions are formed. Residual stresses that are not released, due to for instance the component geometry, can lead to an apparent strength reduction or premature failure of the component. To deal with shape distortion of a part with high shape tolerances, such as an aircraft component, the manufacturers have to compensate the geometry of the mould to obtain a component with the right shape after manufacturing. Today the compensation is made by experienced personnel who rely on thumb rules and previous experience with similar components, materials and manufacturing processes. However, when a new component is to be manufactured for the first time, it is common that the mould needs several modifications before the specifications are met. This is especially true if the component has a complex geometry, different from previously manufactured components, or when new materials are considered. Therefore there is a great interest to increase the ability to understand and predict development of shape distortion and residual stresses. Development of adequate tools for prediction of shape distortions is considered as an important step towards a more cost efficient development, improved performance and optimised manufacturing of composites. To be able to perform shape distortion and residual stress analysis a material model is required. The models ability to describe the behaviour of the polymer matrix during cure is crucial for the accuracy of the analysis. The material behaviour that the model must describe is best explained by considering neat resin cure. During cure the polymer transforms from a liquid to a rubber like solid, which finally transforms to a solid in the glassy state [1]. This means a dramatic change in mechanical properties. Young’s modulus and the shear modulus are approximately 100 times lower in the rubbery than in the glassy state and the coefficient of thermal expansion is approximately 2 to 3 times higher in the rubbery than in the glassy state [2]. The polymer also undergoes chemical shrinkage during cure due to crosslinking. Much of the shrinkage occurs while the resin is a liquid and therefore (in most cases) without stress development, but shrinkage in the later

2 stages of cure (after gelation) leads to residual stresses or shape distortion. A common approximation is to assume the chemical volumetric shrinkage of the resin to be linearly dependent on degree of cure. If the polymer composite is constrained by the mould during cure, shape distortions are formed by release of residual stresses during demoulding. On the other hand, if the composite is free to move during cure, shape distortions may be formed directly without development of macroscopic residual stresses, at least for single curved geometries. The amount of shape distortion after cure in these two cases may differ even for identical cure cycles. The free part is expected to have a higher amount of shape distortions than the constrained. This is because deformations might be frozen into the part when the material transforms from the rubbery to the glassy state (vitrification) during constrained cure (due to dramatically increasing relaxation times during vitrification in the case of a viscoelastic material). Further, when the constrained part is released by demoulding and heated close to or above its glass transition temperature (transformation from glassy to rubbery state) the frozen-in deformations are totally or partly recovered with a corresponding increase in shape distortions. This mechanism has been observed by Kolkman [3] and discussed by Kominar [4]. The simplest and most common residual stress or shape distortion analysis is to assume that the composite is stress free at the cure temperature and then residual stresses are formed during the cooling from the cure temperature to room temperature [5, 6]. In this kind of analysis only thermal effects are accounted for and chemical shrinkage is neglected. This type of analysis may seem reasonable since the rubbery modulus is very low, which for one and two dimensional stress states results in negligible stress development during cure [7]. This is however not true for cases where the polymer is subjected to a three- dimensional stress state, such as laminated composites of complex shape. Several authors have shown, e.g. [8] that the bulk modulus is of the same order of magnitude in the rubbery and the glassy state. This means that significant stresses may develop due to chemical shrinkage also in the rubbery state. A more accurate analysis is to consider residual stress development during the whole cure cycle [9-20]. To obtain good results it is important that the mechanical constitutive model used in the process model is able to capture the mechanisms relevant for residual stress and shape distortion formation. Mechanical constitutive models that have been used for residual stress calculation include both elasticity [9, 10, 17, 19] and viscoelasticity [11-13, 15, 16, 20]. An elastic model manages to handle the glassy- rubbery transition in a realistic way but not the rubber-glass transition. In contrary, an incremental elastic model, manages to handle a rubber to glass but

3 not a glass to rubber transition. Holmberg [19] used an elastic relation and introduced a plastic strain term to handle the rubber to glass transition in a realistic manner. Others have used incremental elastic relations for problems that do not involve glass to rubber transitions, for instance Bogetti and Gillespie [9], Huang et al. [10] and Johnston et al. [17]. The more advance descriptions of a thermoset polymer during cure are based on viscoelasticity. For instance Kim and White [12] and White and Kim [13] showed that the behaviour of neat resin respectively laminates of Hercules 3501- 6 during cure is viscoelastic and thermorheologically complex. Others have approximated the visco-elastic behaviour as thermorheologically simple e.g. [14-16, 20]. The viscoelastic behaviour during cure and in particular thermorheologically complex behaviour is difficult to model accurately for a composite. Other drawbacks are that use of anisotropic viscoelastic models demands extensive material characterisation and for numerical simulations they lead to long calculation times and require large memory for storage of internal state variables. In a previous experimental work by the authors [21], the influence of the cure schedule on spring-in of homogenous balanced laminates was investigated. Resin Transfer Moulding (RTM) was used to manufacture the laminates at conditions typical for high-performance components, that is modest heating rate, long cure time and modest cooling rate. It was demonstrated that, for this process, models for residual stress development and shape distortions must account for thermal expansion (different in glassy and rubbery state), chemical shrinkage due to the crosslinking reaction and frozen-in deformations. An important observation was that, within the range tested, the rate of cooling from in-mould cure temperature to room temperature did not affect the shape distortion. This implies that a mechanical constitutive relation without rate dependence should be sufficient for prediction of residual stresses and shape distortion of composites manufactured by this process So far is it clear that to be able to perform simulations of residual stresses and shape distortions the whole cure schedule have to be considered and the mechanical material model have to be able to model the material behaviour during curing in a realistic manner. Another aspect is to enable simulation of curing and residual stress formation of complex structures with arbitrary geometry and boundary conditions, which can be achieved in a general purpose FE-program. In the literature there are several examples of FE-based process models for prediction of residual stresses. For instance Huang et al. [10] used a cure dependent textile unit cell model in combination with ABAQUS to predict process induced stresses for woven fabric composite structures. Also Poon et al.

4 [15] used ABAQUS to analyse curing of two different components by implementing a viscoelastic temperature and cure dependent material model. The present paper presents a process model for prediction of shape distortions during composite manufacture, with particular emphasis on development and FE-implementation of a simplified mechanical constitutive model that accounts for the mechanisms mentioned earlier. The paper is the first of two companion paper where the second paper [22] is attributed to validation of the process model against experimental results for a particular material system and manufacturing process.

Theory

The focus of the present paper is a new simplified mechanical constitutive model, which will be derived and presented in detail. However, the cure process contains both a heat transfer and a mechanical analysis. For that reason this section will start with presenting the complete thermo-mechanical initial- boundary value problem. This is followed by a detailed presentation of the new mechanical constitutive model. Initial-boundary value problem (IBVP)

The problem is posed using small displacement quasi-static theory for a body consisting of homogeneous anisotropic material with volume Ω and boundary ∂Ω . For a fibrous composite this means that we either have to use homogenised properties or model the fibres and polymer matrix separately.

Thermal IBVP Conservation of energy:

∂⋅()cT ∂ ρ ⋅=−+Qq ∂∂i (1) txi where ρ is density, c specific heat capacity, T temperature, Qi heat flux and q internal heat generation rate (energy released by the resin during the cure process). Thermal constitutive relation (Fourier’s heat law): ∂T Qk=− iij∂ (2) x j = where kijxyzij ,( , , , ) denotes the tensor of thermal heat conductivity. Internal heat generation:

5 =⋅ρ ⋅⋅−dX qHmtot(1 V f ) (3) dt ρ where m, H tot ,Vf and dX dt is density of the matrix, total heat released by the matrix during the curing reaction (energy per unit mass), fibre volume fraction respectively the chemical reaction rate Cure kinetics: dX = fXT(,) (4) dt where X is degree of cure. By definition X is bounded by X ∈[]0,1 (uncured and fully cured resin). Suitable forms of the function f for a particular resin can be found in literature, see e.g. Ref (23). Initial conditions: ()=Ω () Txii,0 T0 x in (5a)

()=Ω () Xxii,0 X0 x in (5b) Boundary conditions: ()=∂Ωψ () () Txtii,,on xt T1 t (6a)

∂Txt(), nxtti =∂Ωφ (),on () jiT∂ 2 (6b) x j where ψ and φ are functions describing the boundary conditions and nj is the ∂Ω ∂Ω unit normal vector to the surface. T1 and T 2 are the temperature and heat flux boundary, respectively. Eqns. (1) to (6) constitute a well-posed thermal IBVP if there is no coupling to the mechanical IBVP. Couplings may be present through stress dependence for the thermal properties λij and c and the cure kinetics and finally stress or displacement dependence for the thermal boundary conditions. Because this present paper focus on the mechanical model the thermal IBVP will not be discussed further.

Mechanical IBVP Conservation of linear momentum (in absence of body forces and inertia effects):

6 ∂σ ij = 0 ∂ (7) x j where σij is the stress tensor that is symmetric in absence of body moments. Strain-displacement relation:

1 ∂u ∂u ε =+i j ij ∂∂ (8) 2 xxji where εij is the Cauchy infinitesimal strain tensor and ui is the displacement vector. Expansional strains:

εεεETC=+ ij ij ij (9) Thermal strains:

t ∂T εαT = (,TX ) dt′ ij∫ ij ∂ ′ (10a) 0 t where the instantaneous coefficients of thermal expansion αij depend on temperature and degree of cure as,

α l ,andXX<≥ TTX()  ij gel g αα=≥ r ≥() ij ij,andXX gel TTX g (10b) α g < ()  ij, TTX g

α l α r α g ij , ij and ij are linear coefficient of thermal expansion in the liquid, rubbery and glassy state, respectively. Tg is the glass transition temperatures that for thermoset resins depend on degree of cure X. Xgel denote degree of cure at gelation. Chemical shrinkage strains: The free chemical shrinkage of neat resinisassumedproportionaltodegreeof cure. However, during free chemical shrinkage of a composite the fibres will impose a constraint on the resin. This leads to different chemical shrinkage in the rubbery and glassy state since the rubbery and glassy modulus of the resin is significantly different. In accordance with this we define the chemical shrinkage as,

t ∂X εβC = (,TX ) dt′ ij∫ ij ∂ ′ (11a) 0 t

7 where the instantaneous coefficients of chemical shrinkage βij depend on temperature and degree of cure as,

β l ,andXX<≥ TTX()  ij gel g ββ=≥ r ≥() ij ij,andXX gel TTX g (11b) β g < ()  ij,TTX g

β l β r β g where i , i and i are the linear coefficient of chemical shrinkage in the liquid, rubbery and glassy state, respectively. Glass transition temperature: TT− λ ⋅ gg0 = X −−−⋅λ (12) TTgg∞ 0 1(1) X whereTg0 and Tg∞ are the glass transition temperature of the uncured (X =0) respectively fully cured system (X =1)and λ a material constant. This is the DiBenedetto equation [24] for instance used by Simon et al. [25], but also other expressions have been used in the literature to relate glass transition temperature to degree of cure. Initial conditions: σ ()=Ω ijx k ,0 0 in (13a)

()=Ω uxik,0 0 in (13b) Boundary conditions: =∂Ω() uxtij(),,on u i01 () xt j u t (14a)

σ ()=∂Ω () () nxtFxtjjik,,on ik u2 t (14b) ∂Ω where ui0 and Fi are functions describing the boundary conditions. u1 and ∂Ω u2 are the displacements and traction boundary, respectively. Equations (7) to (12) subject to the conditions (13-14) are 28 equations that contain 34 unknowns σ ε εεεETC ( ij, ij, ui, ij,, ij ij and Tg). To produce a well-posed mechanical IBVP we need to define a mechanical constitutive relation and this is made in the next section. The mechanical IBVP is coupled to the thermal IBVP through the expansional strains, Eqns. (9-11), and as will be shown in the next section, through the mechanical constitutive relation.

8 Mechanical constitutive relation

Non-linear anisotropic viscoelasticity is a general description that promises potential to accurately describe the behaviour of neat resin and composites during the cure process. However, the experimental data [21] discussed in the introduction suggest that, for some materials under certain process conditions, it is not necessary to capture the details of the rate dependence present in viscoelasticity. To derive a suitable, simplified, constitutive model we will use viscoelasticity as a starting point. It is possible to derive the same mechanical constitutive model using other starting points but here we used viscoelasticity for the following reasons, • to demonstrate that the model is a limiting case of viscoelasticity. This makes it easier to assess whether the simplified constitutive model is valid for a particular case. • to indicate that future generalisations, if necessary, can be made using non-linear and/or thermorheologically complex viscoelasticity as a starting point. • a good approach for incrementalisation is available in the literature [25]. Linear thermo-viscoelasticity for anisotropic and thermo-chemo- rheologically simple materials can be written in integral form as [12],

t ∂−()εεE σξξτ()tC=− (’ ) kl kl d ij∫ ijkl ∂τ (15) 0 where Cijkl is the relaxation modulus tensor. The symbol ξ in Eqn. (15) is reduced time defined by,

t τ 11’’ ’ ξ ==∫∫dtand ξτ d (16) 00aaTT where aT is the time-cure-temperature shift factor. A generalised Maxwell model with P Maxwell elements in parallel with a free spring gives the relaxation () modulus, Ctijkl as,

0,XX<  gel  −t = P  Ct()  ∞ ρ p ijkl CCeXX+⋅∑ p ijkl , ≥,nosumoni,j,k,l (17)  ijkl ijkl gel  p=1 

9 ∞ p ρ p where Cijkl is the fully relaxed modulus matrix, Cijkl and ijkl are spring constants and relaxation times, respectively. To obtain a material model that is simple but still capture most of the important mechanisms for a curing problem we approximate the time-cure- temperature shift factor in the following manner,

ω ,TTX≥ ()  g =  aT lim 1 (18) ω→0 ,TTX< () ω g where Tg is the glass transition temperature which is a function of the degree of cure, X. It can be shown that the constitutive relation, Eqns. (15-18), can be simplified to,

 rE⋅−()εε ≥() CTTXijkl kl kl, g σ =  ij gEgrE (19) CCCTTX()()()εε−− − ⋅− εε , <() ijkl kl kl ijkl ijkl kl kl= g  ttvit where tvit is the last time of the last rubber-glass transition (vitrification). This will not be shown here, instead the incremental form of the constitutive relation that corresponds the approximation (19) will be derived in detail in the next section.

Incrementalisation The formulation of Eqn.’s (15-17) is well suited for incrementalisation and numerical implementation into a general purpose FE-package such as ABAQUS. Following the procedure of Zocher et al. [26], the stresses at time tt+∆ is defined as, σσσ()()+∆ = +∆ ijtt ij t ij (20) ∆σ where the stress increment, ij is approximated by,

∆=∆⋅∆−σεεσER +∆ ijC ijkl() kl kl ij (21)

R where ∆Cijkl and ∆σij are given by,

−∆ξ P  ∞ 1 ρ p ∆=CC +∑ ρ pp ⋅⋅− C1 eijkl ijkl ijkl∆ξ ijkl ijkl , no sum on i, j, k, l (22) p=1 

10 −∆ξ 33P  ρ p ∆=−σ Rp∑∑∑1() −eStijkl ⋅ ij ijkl ,nosumoni, j, k, l (23) klp==11 = 1

p where Sijkl are history dependent state variables defined by a recursive relation as,

−∆ξ ∆−εεE  −∆ξ ρρp ()kl kl p Stpppp()+∆ t = eijkl ⋅ St () +ρ ⋅ ⋅ C ⋅ 1 − eijkl , no sum on i, j, k, l (24) ijkl ijkl ijkl∆ξ ijkl   The reduced time increment is given by, ∆t ∆=ξ (25) aT Zocher et al. [26] showed that this incrementalisation gives exact results when the strain has linear dependence on reduced time within each time-step. Inserting Eqns. (18) into (25) and using a McLaurin expansion of exponential −∆t 1 ρ terms of the form eω ijkl ,

2 −∆t  1 ρ  ijkl ∆∆tt eOω =−1 + ,(26) 11ρρ ijkl ijkl ωω into Eqns. (22) to (24) and performing the limit ω 0gives,

CTTX∞ , ≥ ()  ijkl g ∆=C  P ijkl ∞ +

−≥StI () , T T() X ∆=σ R  ij g ij < () (28) 0,TTXg

0,TTX≥ ()  g StI ()+∆ t = P ij Ip+⋅∆−<εε E () (29) Stij()∑ C ijkl() kl kl , T T g X  p=1

I where a new state variable Sij was defined by,

11 33P Ip≡ Stij()∑∑∑ S ijkl () t (30) klp==11 = 1 Eqn. (27) and (28) into (21) finally yields,

CStTTX∞ ⋅∆()εε −EI −() , ≥ ()  ijkl kl kl ij g ∆=σ P ij ∞ +⋅∆−

∞ Equating the fully relaxed stiffness tensor, Cijkl , with the rubbery modulus tensor, P r ∞ + p Cijkl , and the unrelaxed stiffness tensor, CCijkl∑ ijkl , with the glassy modulus p=1 g tensor, Cijkl , permits us to rewrite Eqns. (29 and 31) with these parameters,

≥() 0,TTXg StI ()+∆ t = ij Igr+−⋅∆−εε E <() (32) Stij()()() C ijkl C ijkl kl kl , T T g X

 rEI⋅∆εε − − ≥ () CStTTXijkl() kl kl ij() , g ∆=σ ij  (33) gE⋅∆()εε − < () CTTXijkl kl kl, g

We now have an incremental constitutive model based on linear viscoelasticity, but with the rate dependence replaced by a path dependence on the state variables εkl, T and X as intended. For the simplified constitutive model to be valid, the following requirements must be put on the material behaviour and manufacturing process: • Neither the glassy nor the rubbery coefficients of thermal expansion depend on degree of cure. • The material behaves linearly elastic within the glassy and the rubbery state; neither the glassy nor the rubbery stiffness depends on temperature or degree of cure. • Heating through the glass-rubber transition is rapid enough and cooling through the rubber-glass transition slow enough to exclude explicit time dependence (rate dependence). Incrementalisation

The mechanical constitutive relation was incrementalised in the previous section. To obtain a FE-based process simulation tool we also need to incrementalise and implement the expressions for expansional strains, Eqns. (9-

12 11), since they are not available in this form in general purpose FE-programmes. Using the Euler forward scheme we obtain,

εεεEEE()()+∆ = +∆ kltt kl t kl (34) where

∆=∆+∆εεεETC kl kl kl ,(35)

∆=εαT ⋅∆ kl kl T (36) and

∆=εβC ⋅∆ kl kl X (37) where ∆T and ∆X are the temperature respectively degree of cure increment over the time step. In the present implementation the coefficients for linear shrinkage α l β l in the liquid state, kl and kl of Eqns. (10b) and (11b), are taken as zero. For some problems this simplification may not be valid.

Model verification

Finite element implementation

The mechanical constitutive model, Eqns. (32-33) presented in the previous section has been implemented in ABAQUS [27] as a user subroutine, UMAT. A user subroutine UEXPAN in ABAQUS was written to treat expansional strains specified as functions of temperature and a field variable that represent the degree of cure, Eqns. (34-37). It is interesting to compare the number of internal state variables that need to be used for full viscoelasticity, Eqn. (24), and for the simplified model, Eqn. (32), the relation between the state variables is shown by Eqn. (30). To accurately describe a realistic transition region we may need around 9 terms (P = 9) in the expression for the time dependent relaxation modulus tensor, Eqn. (17) [13]. Considering an anisotropic material the viscoelastic state variable p 4 Stijkl () has 3 x 9 = 729 components that has to be stored for each integration I 2 point. For the simplified material model the state variable, Stij (),hasonly3 =9 components. Considering the symmetry of the terms in the stiffness tensor the relation is 324 versus 6 components so the difference is obvious. Of course, generally a fully viscoelasticity model is more accurate than the simplified model presented here. But for the material, cure schedule and manufacturing

13 process presented in Ref. (21) the simplified model appears accurate enough. Validation of the capability to accurately predict shape distortions for a particular material system and manufacturing process [21] is presented in the companion paper [22]. Test cases

Several test cases have been used to verify that the ABAQUS implementation of the process model has been properly made. The implementation of the thermal problem, including cure kinetics, follows what has been previously presented by other authors [10, 15, 28] and has been verified against a separate in-house 1- dimensional cure simulation software [29] for isothermal, constant heating rate and adiabatic conditions. Here we will focus on verification of the mechanical constitutive model and results for two one-dimensional test cases that provide some insight into the behaviour of the proposed mechanical constitutive relation are presented. In the analysis we have used mechanical properties of the epoxy system LY5052 [30], see Table 1.

Table 1. Properties of LY5052 epoxy matrix. Property Source Symbol Value Unit Young’s modulus (glassy state) * Eg 2.6 GPa Poisson’s ratio (glassy state) * νg 0.38 - g Shear modulus (glassy state) E gg()2(1⋅+υ ) G 0.94 GPa Coeff. of thermal expan. (glassy state) [30] αg 71 10-6/°C Young’s modulus (rubbery state) Gr⋅2⋅(1+νr) Er 0.028 GPa Poisson’s ratio (rubbery state) ** νr 0.497 - Shear modulus (rubbery state) Gg /100** Gr 0.0094 GPa Coeff. of thermal expan. (rubbery state) 2.5⋅αg** αr 178 10-6/°C Total volumetric chemical shrinkage * ∆V -7 % **** ∆ β − Coefficient of chemical shrinkage V 3 i 73 % Degree of cure at gelation ** χgel 0.50 - ° Tg at X = 0 (Eqn. 30) *** Tg0 -41 C ° Tg at X = 1 (Eqn. 30) *** Tg∞ 136 C Material constant in Eqn. 30 λ 0.44 - *: Measured value. **: Assumed value. ***: Determined from DSC data. ****: For a neat resin the coefficient of chemical shrinkage is considered the same in both the glassy and rubbery state.

In the first case we consider a fully cured rod of neat resin subjected to simultaneous temperature changes and mechanical load. In the second case a curing rod of neat resin with constrained ends is considered, see Figure 1. The two cases were modelled in ABAQUS using ten three dimensional brick element

14 with 8 nodes called C3D8 [27]. The temperature and degree of cure were considered as known and prescribed as predefined field variables.

σ t<150s t > 150s Case 1 Case 2 Figure 1. Test cases.

Case 1 As discussed before the rate dependence is replaced by a path dependence on the state variables mechanical strain, temperature and degree of cure. The resin exhibits its unrelaxed glassy modulus in the glassy state and the fully relaxed modulus in the rubbery state. Figure 2 shows a) applied temperatures, b) applied stress and finally, c) predicted expansional, mechanical and total strain in the rod. The rod is considered fully cured with a glass transition temperature, Tg,of 136°C according Table 1. At the start the rod is at a temperature of 20°Cand during the following 100 seconds the rod is heated to 160°C. After 83 seconds the rod reaches its glass transition temperature, Tg and transforms from a glassy solid into a rubber like solid. The transformation is approximated as a step change of the mechanical properties and can be observed in the strain plot where the coefficient of thermal expansion is changed. Between 100 and 200 seconds the temperature is kept constant at 160°C but the rod is now subjected to mechanical loading in rubbery state, where the stiffness is low and the polymer is in thermodynamic equilibrium. During the following 100 seconds up to 300 seconds the rod is cooled to 20°C with constant mechanical load. Also here there is a transition of the polymer but this time from the rubbery state into the glassy state where the stiffness is much higher. In the glassy state, the polymer is no longer in thermodynamic equilibrium, which results in an increasing difference between the actual and equilibrium deformation. The rod is then kept at constant temperature, but this time at 20°C and the mechanical load is removed. During unloading the strain decrease but the decrease is almost not visible in the strain plot, because the stiffness of the material is much higher now compared to when

15 the rod was loaded in rubbery state. For this reason the rod do not return to the zero strain configuration after unloading, even if the rod is at the same temperature and degree of cure as at the start. The remaining strain is denoted as frozen-in strain since it is created at the transition from rubbery to glassy state in a mechanically loaded specimen. The frozen deformations are released the next time the material is heated above the glass transition temperature, which is shown in Figure 2 at 483 seconds. The polymer then goes back to equilibrium and the frozen-in strain is recovered, which results in a step change of strain in the unloaded rod. Finally the specimen is cooled from 160°Cto20°Candthe original configuration of zero strain is recovered. The comparison between analytic calculations and ABAQUS results shows no difference.

160

120

80

Temperature (°C) 40

0 0 100 200 300 400 500 600 time (s) T Tg

600000

500000

400000 (Pa)

V 300000

Stress 200000

100000

0 0 100 200 300 400 500 600 time (s)

3.5%

3.0%

2.5%

H 2.0%

1.5% Strain 1.0%

0.5%

0.0% 0 100 200 300 400 500 600 time (s)

Mechanical Expansion Total Figure 2. Test case 1.

16 Case2 In the second case the rod is constrained in both ends according to Figure 1. In this case there is no mechanical loading but the rod is curing and stresses formed due to chemical and thermal expansional strains. This case is close to what actually happens when a polymer or polymer composite is cured in a rigid mould, but here only one dimension is considered. Figure 3 presents a) applied temperature and degree of cure b) stresses and finally c) expensional, mechanical and total strain. During the first 100 seconds the temperature is constant at 120°C and, in this example, the degree of cure considered to increase linearly with time. At 50 seconds the degree of cure is 0.5, which is the gelpoint according to Table 1, and the polymer transforms from a liquid to a rubber like solid. From this point stresses start to form, which is shown in the stress plot but due to the low stiffness in rubbery state the stress is quite small at this stage. This should not be interpreted as if the chemical shrinkage in rubbery state always leads to negligible stresses. For three-dimensionally constrained load cases stress development is proportional to the bulk modulus and the bulk modulus is, in contrast to Young’s modulus and shear modulus, of the same order of magnitude in the glassy and rubbery state [8].

When the degree of cure increases the glass transition temperature, Tg,also increases according to Eqn. (30). At 96 seconds Tg reaches the cure temperature and vitrification occurs, the polymer transforms from a rubber like solid to a glassy solid. The chemical shrinkage continues in the glassy state where the stiffness is much higher which is observed in the stress plot, the stress increases faster from 96 seconds. After vitrification the polymer is no longer in thermodynamic equilibrium and the difference between the actual and equilibrium deformation increase with increasing stress. The rod has now reached its final degree of cure and the glass transition temperature is 136°C according to Eqn. (30) and Table 1. From 100 seconds until 150 seconds the temperature decrease with constant rate from 120°Cto20°C,allthewayinthe glassy state. At 150 seconds the boundary conditions are changed and one of the constrained ends is released and become free to move, see Figure 1. The stresses are then released and deformations are formed in the same way as when a component is demoulded during composite manufacturing. Because the polymer is not in thermodynamic equilibrium there are strains frozen into the material, see the mechanical strain plot in the graph. This strain will be released the next time the rod is heated above Tg. This heating is performed between 150 seconds and 250 seconds and the release of the frozen strains is manifested as a step change of the total strain at 142°C. As for Test case 1 the comparison between analytic calculations and ABAQUS results shows no difference.

17 1.0 160 0.8 120 0.6 80 0.4

Temperature (°C) 40 0.2 Degree of cure (-)

0 0.0 0 50 100 150 200 250 time (s) T Tg Degree of cure

25

20

(MPa) 15 V 10

Stress 5

0 0 50 100 150 200 250 time (s)

2.0%

1.0% H 0.0%

Strain 0 50 100 150 200 250 -1.0%

-2.0% time (s)

Mechanical Expansion Total Figure 3. Test case 2.

Conclusions

A simplified mechanical constitutive model, applicable for a curing resin or homogenised curing composite, was derived in incremental form. The model is derived from of linear viscoelasticity but rate dependence is replaced by a path dependence on the state variables: strain, degree of cure and temperature. The model captures the mechanisms that were identified in a previous paper [21] as important for prediction of shape distortions: thermal expansion, different in glassy and rubbery state, chemical shrinkage and recovery of frozen-in deformations. The model was also implemented in a general purpose FE-program and verified against analytical test cases. As compared to cure simulations using a fully viscoelastic mechanical model the present model lead to significant savings

18 in computational time, memory requirements and costs for material characterisation.

Acknowledgments

This work was performed in collaboration with Saab AB and is a part of a national aeronautic research program (NFFP) funded by the Defence Materials Administration in Sweden (FMV). Funding was also obtained from the SICOMP Foundation and the Swedish Foundation for Strategic Research through the Integral Vehicle Structure research school (IVS). The authors also gratefully acknowledge Professor D. H. Allen (College of Engineering and Technology, University of Nebraska-Lincoln) for valuable and interesting discussions regarding the modelling and advice on the disposition of the paper.

References

1. Aronhime MT, Gillham JK. Time-Temperature-Transformation (TTT) Cure Diagram of Thermosetting Polymeric Systems. In: Dusek K., editor. Epoxy Resins and Composites III, Berlin: Springer-Verlag, 1986:83-113. 2. Shimbo M, Ochi M, Shigeta Y. Shrinkage and Internal Stresses during Curing of Epoxide Resins. Journal of Applied Polymer Science 1981;26:2265-2277. 3. Kolkman J. Testing epoxide castings by thermal analysis. Holectecniek, 1974;4(2):29-35. 4. Kominar V. Thermo-Mechanical Regulation of Residual Stresses in Polymers and Polymer Composites. Journal of Composite Materials 1996;30(3):406- 415. 5. Jun WJ, Hong CS. Cured Shape of Unsymmetrical Laminates with Arbitrary Lay-Up Angles. Journal of Reinforced Plastics Composites 1992;11:1352- 1366. 6. Tseng SC, Osswald TA. Prediction of Shrinkage and Warpage of Fiber Reinforced Thermoset Composite Parts. Journal of Reinforced Plastics Composites 1994;13:698-721. 7. Lange J, Toll S, Månson JAE, Hult A. Residual stress build-up in thermoset films cured above their ultimate glass transition temperature. Polymer 1995;36(16):3135-3141. 8. Plepys AR, Farris RJ. Evolution of residual stresses in three-dimensionally constrained epoxy resin. Polymer 1990;31:1932-1936.

19 9. Bogetti TA, Gillespie JH. Process-Induced Stress and Deformation in Thick- Section Thermoset Composite Laminate. Journal of Composite Materials 1992;26(5):626-660. 10.Huang X, Gillespie JW, Bogetti T. Process induced stress for woven fabric thick section composite structures. Composite Structures 2000;49:303-312. 11.White SR, Hahn HT. Process Modeling of Composite Materials: Residual Stress Development during Cure, Part I. Model Formulation. Journal of Composite Materials, 1992;26:2402-2422. 12.Kim YK, White SR. Stress relaxation of 3501-6 Epoxy Resin During Cure. Polymer Engineering and Science 1996;36(23):2852-2862. 13.White SR, Kim KK. Process-Induced Residual Stress Analysis of AS4/3501- 6 Composite Material. Mechanics of Composite Material and Structures 1998;5:153-186. 14.Zhu Q, Geubelle PH. Dimensional Accuracy of Thermos set Composites: Shape Optimization. Journal of Composite Materials, 2002;36(6):647-672. 15.Poon H, Koric S, Ahmad MF. Towards a Complete Three Dimensional Cure Simulation of Thermosetting Composites. International Conference on advanced Composites 1998. 16.Prasatya P, McKenna GB, Simon SL. A Viscoelastic Model for Predicting Isotropic Residual stresses in Thermosetting Materials: Effects of Processing Parameters. Journal of Composite Materials 2001;35(10):826-848. 17.Johnston A, Vazari R, Poursartip A. A Plane Strain Model for Process- Induced Deformation of Laminated Composite Structures. Journal of Composite Materials 2001;35(16):1435-1469. 18.Fernlund G, Rahman N, Courdji R, Bresslauer M, Poursartip A, Willden K, Nelson K. Experimental and numerical study of the effect of cure cycle, tool surface, geometry, and lay-up on the dimensional fidelity of autoclave- processed composite parts. Composites Part A: Applied Science and Manufacturing 2002;33:341-351. 19.Holmberg JA, Influence of Chemical Shrinkage on Shape Distortion of RTM Composites, Proc 19th International SAMPE European Conference of the Society for the Advancement of Material and Process Engineering, Paris, France, 22-24 April, 1998:621-632. 20.Adolf D, Martin JE. Calculation of stress in Crosslinking Polymers. Journal of Composite Materials 1996;30:13-34. 21.Svanberg JM, Holmberg JA. An experimental investigation on mechanisms for manufacturing induced shape distortion in homogenous balanced laminates. Composites Part A: Applied Science and Manufacturing 2001;32(6):827-838.

20 22.Svanberg JM, Holmberg JA. Predictions of shape distortion, Part II Experimental validation and analysis of boundary conditions. Submitted Composites Part A. 23.Yousefi A, Lafleur PG and Gauvin R. Kinetic Studies of Thermoset Cure Reaction: A Review. Polymer composites 1997;18(2):157-168. 24.Nielsen LE. Cross-Linking-Effects on Physical Properties of Polymers. J. Macromol. Sci. –Revs. Macromol. Chem. 1969;C3(1):69-103. 25.Simon SL, McKenna GB, Sindt O. Modeling the Evolution of the Dynamic Mechanical Properties of a Commercial Epoxy During Cure after Gelation. Journal of Applied Polymer Science 2000;76(4):495-508. 26.Zocher MA, Grooves SE and Allen DH. A Three-Dimensional Finite Element Formulation for Thermoviscoelastic Media. International Journal for Numerical Methods in Engineering 1997;40:2267-2288. 27.Hibbit, Karlsson & Sorensen Inc. ABAQUS/Standard User’s manual, Version 5.8, 1998. 28.Joshi SC, Liu XL, Lam YC. A numerical approach to the modelling of polymer curing in fibre-reinforced composites. Computer Science and Technology 1999;59:1003-1013. 29.Gebart BR. SICOMP/TCV – A suite of computer programs for material characterization and simulation of cure and heat transfer. SICOMP Technical Report 92-008. SICOMP AB, Box 271, SE-941 26 Piteå, Sweden, 1992. 30.Data Sheet: Araldite LY5052/ Hardener HY5052. Ciba Polymers, Switzerland, 1994.

21 Paper C

Prediction of shape distortions

Part II. Experimental validation and analysis of boundary conditions

Paper C Prediction of shape distortions

Part II. Experimental validation and analysis of boundary conditions

J. Magnus Svanberg a, b and J. Anders Holmberg a

aSICOMP AB, Swedish Institute of Composites, Box 271, 941 26 Piteå, Sweden bLuleå University of Technology, Division of Polymer Engineering,971 87 Luleå, Sweden

Abstract

During cure of thermoset composite structures residual stresses and/or shape distortions are always present. Residual stresses can cause apparent strange reduction or failure even prior to demoulding respectively shape distortions can deform a component so that the component is useless. For this reason a lot of effort have been spent to develop simulation tools capable of sufficiently accurate prediction of residual stresses and/or shape distortions to avoid problems in an early stage before the first component is manufactured. In a companion paper a process model for shape distortion predictions was developed and implemented into ABAQUS. In the present paper the model is validated for a material and cure schedule typical for RTM and autoclave processes. Comparisons between predicted and experimental shape distortion shows that the model and simulation approach used capture both effects from different cure schedules as well as the mechanical interaction between composite and tooling during in-mould cure. The results show that changing the mechanical boundary conditions significantly affects the shape distortion prediction. Therefore accurate modelling of the composite-tooling interaction is an important part of a shape distortion analysis.

Keywords: Shape distortion; B. Cure behaviour; A. Finite element analysis (FEA)

1 Introduction

During cure of thermoset composite structures residual stresses and/or shape distortions are always formed. Usually a high performance composite structure iscuredatanelevatedtemperatureandthematerialshrinkageduringcoolingto the room temperature. The thermoset matrix also shrinks chemical due to the crosslinking reaction. The amount of residual stresses and/or shape distortions depend on a number of factors e. g. laminate factors such as lay-up and gradients in fibre fraction or non isothermal curing leading to gradients in degree of cure and temperature. Other factors are the cure schedule and the behaviour of the thermoset resin during cure. Residual stresses can cause apparent strength reduction or failure even prior demoulding respectively shape distortions can deform a component so that the component becomes useless. For this reason a lot of effort have been spent to develop simulation tools to be able to predict residual stresses and/or shape distortions prior to manufacture of the first component. Bogetti and Gillespie [1] presented early a one-dimensional model to investigate the effect of gradients in temperature and degree of cure on residual stresses in thick laminates. The model combined cure kinetics, through- thickness heat conduction and elastic laminated plate theory. Also White and Hahn [2, 3] was early with a model combining cure kinetics with a viscoelastic material behaviour to predict residual stresses in composite materials during cure. Later White and Kim [4] showed that the behaviour of a curing epoxy is rheologically complex and developed a rheologically complex visco-elastic finite element model and investigated curing of a thin rectangular cross ply laminate. More recently Prasatya et. al. [5] used a termo-viscoelastic model to predict isotropic residual stresses in three dimensional thermoset material and showed that an elastic solution overestimated residual stresses and viscoelastic effects can not be neglected. Other has used elastic constitutive models e.g. Johnston et. al. [6] developed a plane strain finite element model using a cure- hardening, instantaneous linear elastic constitutive model to predict process- induced deformations of laminated composite structures. Fernlund et. al. [7] used this model to investigate effects of cure cycles, tool surfaces, geometry and lay-up on shape distortion. For industrial use of cure simulation tools it is often considered as an advantage to use a general purpose FE-program. At least if the industry already uses a general purpose FE-program for other analysis, because of the in-house knowledge of pre and post processing and that sometimes geometry models of components and tools already exists. For instance Poon et al. [8] implemented

2 cure kinetics and a viscoelastic temperature and cure dependent material model in ABAQUS and analysed curing of a bent plate and a hollow cylinder. Huang et al. [9] used a cure dependent textile unit cell model in combination with ABAQUS to predict process induced stresses for woven fabric composite structures. The objective of the present paper is to validate a simplified mechanical constitutive model for prediction of shape distortions that was presented in a companion paper [10]. In the companion paper the simplified mechanical constitutive model was derived and implemented in the general purpose FE- program ABAQUS. In the present paper the FE-based simulation tool is used to produce predictions of spring-in for angle brackets cured at different conditions. The predictions are then compared to experimental results presented in Ref. (11) to validate the model. To obtain a fair comparison between predictions and experiments it is important that proper mechanical boundary conditions are used during in-mould cure and the results is discussed in relation to the experimental observations. The first part of the present paper, sections 2 and 3 presents the materials used and material characterisation performed to obtain the material properties needed for the simulations. For completeness the shape distortion experiments, previously presented in Ref. (11) are briefly described in section 4. In Sections 5 and 6 the FE-models are described and the simulation results discussed, including the use of different mechanical boundary conditions. Then simulation results are compared with experimental results in Section 7. Section 8 includes simulations and a discussion of the effect of performing the post cure in the mould and finally section 9 concludes the paper.

Materials

The epoxy system ®Araldite LY5052 / Hardener HY5052 [12], which is suitable for resin transfer moulding at a wide range of in-mould cure temperatures, and a 300 g/m2 E-glass weave, Hexcel 7781-127 [13] was used in this study.

Material characterisation

To be able to perform a cure simulation including residual stress development and shape distortion, using the process model and simulation tool described in Ref. (10) a number of material properties are needed. The determination of those properties is discussed in the following subsections.

3 Matrix

Mechanical properties In the present simplified mechanical model [10] the mechanical properties are assumed to be constant within each material phase. There are one set of properties in the rubbery state and another set of properties in the glassy state, respectively. In the glassy state the mechanical properties of the isotropic matrix can be found by mechanical testing at room temperature. For instance if Young’s g ν g g modulus, Em and Poisson ratio, m is measured the shear modulus, Gm and bulk g modulus, K m are calculated by,

E g G g = m m ⋅+ν g (1) 2(1m )

E g K g = m m ⋅−⋅ν g (2) 3(12m ) Subscript m and superscript g denote matrix and glassy state, respectively. From the glassy state properties the rubbery state properties can be calculated based on following assumptions: the coefficient of linear thermal expansion is approximately 2.5 times higher in rubbery than in glassy state [14], the coefficient of linear thermal expansion multiplied with the bulk modulus is approximately equal in glassy and rubbery state [15] and finally the shear modulus in rubbery state is approximately 1% of the glassy shear modulus.

ααrg≈⋅ mm2.5 (3)

α g KKrg≈⋅m mmα r (4) m

G g r ≈ m Gm (5) 100

⋅−⋅rr ν r = 32KGmm m ⋅+⋅rr (6) 62KGmm

rrr=⋅ +υ ⋅ EGmmm2(1 ) (7)

4 where α and υ is the coefficient of thermal expansion respectively Poisson ratio. Superscript r means rubbery state.

Results Five specimens of fully cured neat resin were tested in a test machine (INSTRON 8501/H0162) according to ASTM D3039. On three specimens longitudinal strain was measured using an extensiometer. On two specimens both longitudinal and transverse strains were measured with strain gauges. Consequently, the tangent Young’s modulus was averaged from five specimens and tangent Poisson ratio was calculated from two specimens. The result is shown in Table 1. The mechanical properties in the rubbery state presented in Table 2 are calculated using Eqn. (1-7). The glassy coefficient of thermal expansion is taken from the material suppliers data sheet [12].

Table 1. Mechanical properties of LY5052 in the glassy state. Property Symbol Unit Source Value

g a. Young’s modulus Em GPa 2.6 ν g a. Poisson ratio m --- 0.38 α g -6 ° Coeff. of thermal expansion m 10 / C [12] 71 g Shear modulus Gm GPa Eqn. (1) 0.94 g Bulk modulus K m GPa Eqn. (2) 3.6 a. Measured by the authors.

Table 2. Mechanical properties of LY5052 in the rubbery state. Property Symbol Unit Source Value α r -6 ° Coeff. of thermal expansion m 10 / C Eqn. (3) 178 r Bulk modulus K m GPa Eqn. (4) 1.4 r Shear modulus Gm GPa Eqn. (5) 0.0094 ν r Poisson ratio m --- Eqn. (6) 0.497 r Young’s modulus Em GPa Eqn. (7) 0.0028

Relation between glass transition temperature and degree of cure The mechanical properties are strongly related to the material state. The material is in the rubbery state when the temperature is above the glass transition temperature, Tg, or vice versa when the temperature is below Tg the material is in the glassy state. For a thermoset resin Tg is related to the degree of cure, X and for the modelling purpose of the present paper this relation has to be

5 experimentally determined. A common description of this relationship is the DiBenedetto [16] equation, TT− λ ⋅ gg0 = X −−−⋅λ (8) TTgg∞ 0 1(1) X λ used by for instance Simon et. al. [17]. Tg0 , Tg∞ and is the glass transition temperature prior to cure, the glass transition temperature of the fully cured system (X = 1) and a material constant, respectively. The constants in the Dibenedetto equation were found by curing samples of neat resin at different temperatures for appropriately long times to obtain samples with different degree of cure. These samples were then analysed by a DSC

(Differential Scanning Calorimetry) where Tg and the residual heat of the ∆ reaction, Hr was measured from dynamic scans [18] at a scaning rate of

40°C/min. Tg was taken as the point of the curve where the specific heat change is half of the change in the complete transition. The degree of cure for a particular sample is given by, ∆−∆HH X = total r ∆ (9) Htotal ∆ where Htotal is the total heat of reaction released from the material, which is measured by a DSC scan of an uncured sample. The Dibenedetto equation is then established by fitting the equation to the degree of cure and corresponding glass transition temperature of the different samples.

Results Figure 1 presents glass transition temperature versus degree of cure for seven different samples cured isothermal in an oven at different temperatures and curing times. The line in Figure 1 represents the Dibenedetto, Eqn. (8) where the λ parameters Tg0 , Tg∞ and are –41, 136 respectively 0.44 as determined by a least square fit to the experimental values.

6 140 120 100 80 60 40 Tg (°C) 20 0 -20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 -40 Degree of cure, X

100C/22h 80C/24h 60C/66h 40C/96h 60C/45min 60C/60min 60C/90min DiBenedetto model

Figure 1. Glass transition temperature versus degree of cure, neat resin.

Chemical shrinkage

Method Figure 2 illustrates the density change during isothermal cure of a thermoset [14]. Point A corresponds to the density at room temperature of the uncured liquid resin. Most epoxy resins are cured at an elevated temperature and in this study the angle brackets to be discussed in section 4 were in-mould cured between 40 and 120°C[11]. The line between point A and B in Figure 2 illustrates heating of the liquid resin to the cure temperature. The chemical reaction, the cross linking, is in this illustration considered to start at point B and finish at point C. During cure the resin goes from a liquid state through a rubbery state to a glassy state where the reaction slows down and finally stops. After finished crosslinking the solid material is cooled to room temperature and point D corresponds to the density of the fully cured material at room temperature. Because the coefficient of thermal expansion is different in the liquid state compared to the solid state [14] the slope of line A-B is steeper than line C-D in Figure 2. This means that the shrinkage at room temperature (difference between point A and D) contains both a thermal and a chemical effect. The shrinkage at the cure temperature (difference between point B and C) on the other hand is the isolated chemical

7 shrinkage, in other words to be able to measure chemical shrinkage the measurement has to be performed isothermally at the cure temperature. An experiment was designed to measure the chemical shrinkage of the resin system LY5052/HY5052 [12]. A gaspycnometer (Accupyc 1330 equipment from Micrometrics) was used to measure density. This equipment does not have a temperature control system and therefore the measurements are carried out with the equipment at ambient temperature, which results in a measurment temperature around 30°C. The particular epoxy does not reach 100% degree of cure at this cure temperature, and as a consequence the total chemical cure shrinkage cannot be measured directly. However, if the shrinkage is assumed proportional to the degree of cure and the degree of cure after 30°Ccuringis known the total cure shrinkage can be estimated from the experiment. The total ∆ c cure shrinkage, Vm is given by,

ρρliquid()TT− cured () 1 ∆=V c mcuremcure ⋅ m ρ cured (11) m()TXT cure () cure Resin plates were prepared by isothermally curing 2mm thick and ∅100mm resin samples on a steel plate. The curing was performed in an oven at 30°Cfor 25 hours. After the curing the plates were cut into smaller specimens. The density at 30°C of the cured specimens and uncured liquid resin was then determined in the gaspycnometer, followed by measurements of the glass transition temperature using DSC. Using Eqn. (8) and the result from the glass transition measurements we obtain an estimate of the degree of cure. Together with the degree of cure, density of the cured and liquid resin the total chemical shrinkage can be assessed by Eqn. (11).

Density

D

C A

B

RT Cure Temperature temperature Figure 2. Illustration of density change during curing.

8 Results The calculations and results from the chemical shrinkage experiment are presented in Table 3.

Table 3. Chemical shrinkage (Cure temperature 30°C). Property Symbol Unit Source Value ρ liquid 3a. Density at 30°Cand χ = 0 m (T) kg/m 1.099 ° ° ρ cured 3a. Density at 30 C after cure at 30 Cfor25h m (T ) kg/m 1.157 b. Degree of cure after cure at 30°C for 25h X 30°Ch / 25 --- 0.73 ∆ c Total volumetric chemical shrinkage Vm % Eqn. (11) -7 εc ε cc=∆ Total linear chemical shrinkage % Vm 3 -2.33 a. Measured at SICOMP. b. Glass transition temperature measured by DSC and the degree of cure calculated using Eqn. (8).

Laminate properties

Mechanical properties

Based of the fibre fraction, the lay-up (0/90)16 and the mechanical properties of fibre and matrix the mechanical properties of the laminate (both in the glassy and the rubbery state) were calculated using self-consistent-field micro- mechanics [19] and 3D-laminate theory [20], see Table 4 and Table 5 respectively. The average fibre content in the manufactured angle brackets was 49% by volume as determined by pyrolysis (560°C for 4h). Mechanical properties of E- glass fibres were found in Ref. (21); Young’s modulus, Poisson’s ratio and coefficient of thermal expansion are 76 GPa, 0.22 and 4.9⋅10-6 1/°C respectively. To check the precision of the estimated laminate properties the following properties were calculated from the tangent of stress strain curves from tensile tests according ASTM D3039:E11, E22 and ν12. The tensile tests were performed at room temperature, in the glassy state. The results are shown in Table 5 and the three measured mechanical properties agree well with the calculated values.

9 Table 4. Ply properties in glassy and rubbery state. Glassy Rubbery Description Symbol state state

Longitudinal Young’s modulus EL (GPa) 38.6 37.2 Transverse Young’s modulus ET (GPa) 7.0 0.1 In-plane Poisson’s ratio νLT (-) 0.30 0.36 Transverse Poisson’s ratio νTT (-) 0.52 0.98

In-plane shear modulus GLT (Gpa) 2.6 0.03 Transverse shear modulus GTT (Gpa) 2.3 0.03 -6 Longitudinal coeff. of thermal expansion αL (10 /°C) 7.2 5.0 -6 Transverse coeff. of thermal expansion αT (10 /°C) 50.7 136.6 -4 -5 Longitudinal coeff. of chemical shrinkage βL (1 /°C) -8.0⋅10 -9.0⋅10 -2 -2 Transverse coeff. of chemical shrinkage βT (1 /°C) -1.6⋅10 -1.8⋅10

Table 5. Laminate properties of LY5052 / 7781 laminate (Vf=49%) in the glassy state and rubbery state. Calculated Experimental Glassy Rubbery Glassy Description Symbol state state state

In-plane Young’s modulus E11 (GPa) 23.9 19.8 24.8

In-plane Young’s modulus E22 (GPa) 22.0 17.6 22.3

Out-of plane Young’s modulus E33 (GPa) 8.4 2.3 ---

Major Poisson ratio ν12 (-) 0.10 0.002 0.13

Out-of plane Poisson ratio ν13 (-) 0.45 0.84 ---

Out-of plane Poisson ratio ν23 (-) 0.46 0.85 ---

In-plane shear modulus G12 (GPa) 2.55 0.03 ---

Out-of-plane shear modulus G13 (GPa) 2.44 0.03 ---

Out-of-plane shear modulus G23 (GPa) 2.42 0.03 ---

In-plane coefficient of thermal expansion α11 (10-6/°C) 14.5 5.4 ---

In-plane coefficient of thermal expansion α22 (10-6/°C) 15.9 5.5 --

Out-of-plane coeff. of thermal expansion α33 (10-6/°C) 66.4 264.8 --- -3 -5 In-plane coefficient of chemical shrinkage β11 (1 /°C) -3.4⋅10 -7.3⋅10 --- -3 -5 In-plane coefficient of chemical shrinkage β22 (1 /°C) -3.9⋅10 -8.6⋅10 --- Out-of-plane coefficient of chemical β ° ⋅ -2 ⋅ -2 shrinkage 33 (1 / C) -2.2 10 -3.5 10 ---

10 Shape distortion experiments

In a previous paper by the authors [11] experimental data for spring-in of glass- fibre epoxy composites were presented. The experiments were performed with 4 mm thick angle brackets manufactured by RTM, in a steel mould with accurate temperature control. Different in-mould cure temperatures (40, 60, 80 and 100°C) were used to point out and separate different mechanisms responsible for spring-in. All angle brackets were manufactured using a three step cure schedule, described in Table 6, where the two last cure steps are identical for all specimens. The first cure step was performed in the mould and the following two cure steps were made freestanding in an oven, experimental setup 1-4 in Table 7. Note that the temperature of the second and third cure step (120 and 150°C) is significantly lower respectively higher than the ultimate glass transition temperature of the resin. The spring-in was measured in a measurement machine at room temperature after each cure step. Spring-in, ∆θ means the increase of the external angle in a corner of a composite component after curing, see Figure 3.

Figure 3. Distorted angle section. To further investigate the influence of mechanical constraints during in- mould cure two additional angle brackets have been manufactured for the present paper using the same manufacturing procedure as in Ref. (11). The fundamental difference is however that the two additional brackets were in- mould cured during the two first cure steps and only the last cure step was freestanding, experimental setup 5 in Table 7. In all experiments performed in Ref. (11) and the additional experiments presented in this work, the mould surfaces was covered with Mylar films. The primary purpose with the Mylar was to eliminate manufacturing defects caused

11 by laminate shrinkage in the thickness direction in a very rigid mould. The film also reduced the friction/adhesion between the mould and the laminate.

Table 6. Cure schedules. Cure schedule (qC/h) Cure schedule Step 1 Step 2 Step 3 1 40 / 24 120 / 4 150 / 2 2 60 / 20 120 / 4 150 / 2 3 80 / 15 120 / 4 150 / 2 4 100 / 12 120 / 4 150 / 2

Table 7. Cure schedule and experimental boundary conditions. Experimental Cure Step 1 Step 2 Step 3 setup schedule 1 1 In-mould Free Free 2 2 In-mould Free Free 3 3 In-mould Free Free 4 4 In-mould Free Free 5 3 In-mould In-mould Free Degree of cure after in-mould cure of the angle brackets

After the first cure step the glass transition temperature was measured by DSC scans on two angle brackets for each isothermal in-mould cure temperature, the result is shown in Figure 4 with 95% confidence limits calculated from the pooled standard error.

12 140

120

100

80

Tg (°C) 60

40

20

0 20 40 60 80 100 120 Tcure (°C)

Figure 4. Glass transition temperature versus isothermal in-mould temperature, angle brackets. By assuming that the relation between the glass transition temperature and the degree of cure of the matrix is valid also for laminates the degree of cure after the first cure step was calculated from the results in Figure 1 using Eqn. (8), see Table 8. DSC measurements also showed that all specimens were fully cured after the second cure step without dependence on the temperature during the first cure step.

Table 8. Degree of cure after the first cure step, angle brackets. In-mould cure temperature (qC) Degree of cure, Xa. 40 0.78 60 0.88 80 0.92 100 0.95 Calculated by Eqn. (9) and data in Figure 4. Spring-in results

Measured shape distortion after each of the three cure steps is presented in Table 9. For a detailed description of the measurements consult Ref. (11). The nine first specimens were manufactured using experimental setup 1-4 and these results have previously been reported [11]. The last two specimens were manufactured using experimental setup 5.

13 The changes in spring-in from the second to third cure step are presented In Table 10 together with the pooled standard error determined for the average difference in spring-in between cure step 3 and 2. From the pooled standard error, the 95% confidence interval limits were calculated for specimens cured freestanding respectively in-mould cured during second cure step, see Table 11. The result in Table 11 shows that for specimens freestanding during the second cure step there is no change in spring-in between the second and third cure step, because zero is in the interval. This verifies that the specimens were fully cured during the second cure step, as shown by the DSC measurements in the previous section, and that all frozen-in deformations also were released during that step. In-mould curing during the second cure step on the other hand results in a statistically significant increase of the shape distortion during the third freestanding cure step.

Table 9. Measured spring-in, ∆θ Experimental Specimen, Step1, Step2, Step3, a. ∆θ ∆θ ∆θ setup i 1 2 3 1 1 0.53 0.30 0.26 1 8 0.64 0.33 0.35 2 2 0.74 0.74 0.71 2 7 0.81 0.81 0.79 3 3 0.99 1.16 1.14 3 5 1.00 1.19 1.19 3 6 1.02 1.26 1.23 4 4 1.74 2.12 2.09 4 9 1.35 1.74 1.76 5 10 --- 1.08 1.21 5 11 --- 1.26 1.31 a. Experimental setup1–4fromRef.(11)

14 Table 10. Change in spring-in during third cure step for specimens cured with different experimental boundary conditions. Change in spring-in Specimen Experimental (q) a. ϕ BC..=∆θθ −∆ 2..2=−ϕϕBC BC.. i setup B.C. i 32 d ()i 1 1 Free -0.044 5.86⋅10-4 8 1 Free 0.019 1.49⋅10-3 2 2 Free -0.033 1.61⋅10-4 7 2 Free -0.023 8.60⋅10-6 3 3 Free -0.019 6.94⋅10-7 5 3 Free -0.006 1.94⋅10-4 6 3 Free -0.033 1.82⋅10-4 4 4 Free -0.026 4.31⋅10-5 9 4 Free 0.024 1.92⋅10-3 ϕϕFree = Free Average, ()/9∑ i -0.020 10 5 In-mould 0.129 1.56⋅10-3 11 5 In-mould 0.050 1.56⋅10-3 Const. ϕϕ= ()Const. 2 Average, ∑ i 0.090 Sum of squares, ∑ d 2 7.71⋅10-3 2 Pooled Std. Error, sd=−(/(112))∑ 0.029 a. Experimental boundary conditions. For specimens market Free and In-mould the second cure step was performed freestanding respectively in the mould.

Table 11. 95% confidence intervals for ϕ in Table 10. Boundary conditions Free In-mould Number of specimens, n 92 α =− = = t-distribution, 1 95% 5% , t(5% / 2, 11-2) 2.262 Pooled estimate of std. deviation in Table 10, s = 0.029 _ Average, ϕ -0.020 0.090 _ ϕ−⋅ Lower limit, tsn(2.5%,9) / -0.045 0.043 _ ϕ+⋅ Upper limit, tsn(2.5%,9) / 0.005 0.137

FE-model

In the companion paper [10] a process model for cure and shape distortion simulations was implemented into the general purpose FE-package ABAQUS [22], which makes it possible to calculate shape distortion and residual stresses

15 on arbitrary geometries. Here we will present shape distortion simulations of the angle brackets presented in the previous section. The angle brackets were isothermal cured and therefore the temperature and degree of cure is considered spatially uniform throughout the component. This condition implies that the important part of the temperature evolution is known from the cure schedule, Table 6. The degree of cure after each cure step is also known from DSC measurements. Hence prescribed temperature and degree of cure can be used during the calculations. If the angle brackets had not been manufactured at isothermal cure conditions, then cure kinetic simulations would have been necessary to predict the evolution of temperature and degree of cure. To investigate the influence of different mechanical boundary conditions on the spring-in prediction, simulations were performed with three different boundary conditions: 1) Freestanding 2) Fully constrained 3) Contact conditions Freestanding or fully constrained boundary conditions

Figure 5 shows the FE-model of the angle bracket. Here only the angle bracket was modelled and the mould was left out. 6-node bilinear quadrilateral general plane strain elements (CGPE6) were used. Two different mechanical boundary conditions were used during in-mould cure. In the first case the bracket was free to move during the entire cure schedule and in the second case all nodes were fully constrained to obtain a crude approximation of the constraint from the mould. When the angle bracket is considered free to move during the entire cure schedule the mould is assumed to not influence the shape distortion. In reality there is always an effect from the mould and this kind of boundary condition is realistic only if the mould is very soft compared to the composite, and deform with the component. For this freestanding case nodes were only constrained to suppress rigid body motion. The other case when the angle bracket is modelled as fully constrained the component is considered bonded to an absolutely rigid mould during the entire in-mould cure. Constraining all degrees of freedom on all nodes is a quick but approximate way to impose this condition. Removing constraints so that only rigid body motion remains suppressed simulated demoulding.

16 Figure 5. Model of the angle bracket

Contact boundary conditions

Contact boundary conditions are the most realistic but also most expensive to model in-mould cure. In reality the component is neither free to move nor completely stuck to the mould, see Figure 6. In a finite element contact analysis [22], contact between surfaces is treated as constraints that permit forces to be transferred from one component to the other. When the surfaces are separated the constraints are removed and no forces transmitted. In an ABAQUS [22] contact analysis the surfaces that can interact have to be defined in the model as contact pairs. It is also possible to account for friction that constrains relative sliding between two surfaces, but because the mould surfaces were covered with Mylar films during the experiment the contact was assumed to be frictionless in the simulations. To simulate demoulding the model change option in ABAQUS was used to remove the mould elements during the analysis and new constraints were applied to suppress only rigid body motion. Figure 6 shows deformations prior demoulding. The mould and the angle bracket were modelled with 10-node biquadratic general plane strain elements (CGPE10). The model will be used to simulate experimental setup 1-4 in Table 7 where the first cure step is performed in the mould followed by two freestanding post cure steps. The result will be compared to the experimental results and the simulation results obtained using the two simpler mechanical boundary conditions presented in previous section. The model will also be used to

17 simulate the experimental setup 5 in Table 7 where two cure steps were performed in the mould, followed by a third freestanding post cure step.

Figure 6. Mould and angle bracket prior demoulding, symmetry used in the simulations.

Simulation results

Freestanding boundary conditions

When the freestanding boundary conditions are used there will be an isothermal increase in spring-in at the cure temperature due to chemical shrinkage after gelation. This is followed by a further increase in spring-in during cooling to room temperature after complete in-mould cure, as shown in Figure 7 for the 80°C in-mould cure schedule, experimental setup 3 in Table 7. The continuous line represents the first cure step, the in-mould cure, and the dashed line represents the second freestanding cure step. The second cure step starts at room temperature and the glass transition temperature at this point is about 108°C. During the heating to the post cure temperature, 120°C, the spring-in will decrease due to thermal expansion. The heating starts in glassy state and at 108°C the polymer transforms into rubbery state and, as a consequence, the spring-in will decrease faster from this point because of the large through- thickness coefficient of thermal expansion in rubbery state. At 120°Cthe

18 isothermal change in spring-in is an effect of further chemical shrinkage due to further curing during the second cure step. At the end of the second cure step the degree of cure is approximately 100%, which corresponds to a glass transition temperature of 136°C. Consequently, after the isothermal cure the specimen is cooled without transition all the way into the glassy state, which is shown as a straight line in Figure 7. The third cure step is not shown in Figure 7 because the specimen has already reached complete cure during the second cure step and the spring-in will follow the same path during heating and cooling and the spring-in at room temperature remain unchanged.

1.4

1.2

1.0

0.8

0.6 Spring-in (°) 0.4

0.2

0.0 0 20 40 60 80 100 120 140 Temperature (°C)

Cure step1 Cure step2

Figure 7. Predicted spring-in during the first and second cure steps in experimental setup 3, freestanding conditions.

Fully constrained boundary conditions

Spring-in, predicted using fully constrained boundary conditions, versus temperature is shown in Figure 8 for cure conditions according to experimental setup 3. This is the same cure temperature as used to produce Figure 7. The difference is that the angle bracket is considered fully constrained by a rigid mould and there cannot be any shape distortion before demoulding. Instead residual stresses will be formed. The residual stress will be released at demoulding and shape distortions formed, the vertical line at 20°C in Figure 8. The shape distortion after the first cure step is less than for the case where the

19 specimen is considered free to move during the entire cure step, see Figure 7. Thereasonforthedifferenceisdeformations frozen into the specimen due to the constrained in-mould cure. These frozen deformations will be released the first time the specimen transforms to the rubbery state and the polymer softens, which is shown by the vertical jump at the glass transition temperature, at 108°C, in Figure 8. During the second cure step the release of the frozen-in deformations is the only difference between this solution and the solution where the specimen was considered freestanding during both cure steps, discussed in the previous section. The result after the second cure step is the same for both types of simulations.

1.4

1.2

1.0

0.8

0.6 Spring-in (°) 0.4

0.2

0.0 0 20 40 60 80 100 120 140 Temperature (°C)

Cure step1 Cure step2

Figure 8. Predicted spring-in during the first and second cure step for experimental setup 3, fully constrained conditions during cure step 1.

Contact boundary conditions

Figure 9 shows calculated spring-in using the contact boundary condition model of Figure 6. Besides the boundary conditions the simulation is identical to the two previous simulations presented in Figure 7 and Figure 8. The shape of the curve in Figure 9 is almost the same as in Figure 8, except that the contact model allows some deformation during the first cure step, see Figure 6, and a small spring-in is predicted also before demoulding. The spring-in after the vertical jump at 20°C due to demoulding is in the middle of the two previous cases. The vertical jump at the glass transition, 108°C is smaller than for the

20 case of a fully constrained specimen. The reason for these two differences is that the frozen-in deformations are smaller than for a fully constrained specimen.

1.4

1.2

1.0

0.8

0.6 Spring-in (°) 0.4

0.2

0.0 0 20 40 60 80 100 120 140 Temperature (°C)

Cure step1 Cure step2

Figure 9. Predicted spring-in during the first and second cure step in experimental setup 3, contact conditions during cure step 1.

Comparison of simulation and experimental results

In this section the experimental results corresponding to experimental setup 1-4 in Table 7, are compared with simulation results. For these cases the first cure step was performed in the mould and the following two as freestanding. In this case neither the experimental results nor the simulations show any change in spring-in between the second and third cure step. For that reason only the first (in-mould) and the second cure step (freestanding) will be discussed. Shape distortion at room temperature after in-mould cure

After the first cure step (in-mould cure) the predicted spring-in is dependent on the mechanical boundary conditions used in the simulations. Figure 10 shows a comparison between experiential and predicted spring-in at room temperature for the different in-mould cure temperatures, experimental setup 1-4. By considering the predicted spring-in predictions presented in Figure 10 it is clear that the free boundary condition result in largest predicted spring-in after the in-mould cure step. The fully constrained (fixed) boundary conditions result

21 in the smallest spring-in and the contact boundary condition gives a result in- between those two, as expected. The experimental results show close agreement with the predictions obtained using the contact boundary conditions, at least for the specimens in-mould cured at 40, 60 and 80°C respectively, experimental setup 1-3 in Table 7. This is however not the case for specimens manufactured with the 100°C in-mould cure schedule, experimental setup 4, but this was also the experimental setup that showed the largest experimental scatter, specimen 4 and 9 in Table 9. Possibly the poor comparison between predicted spring-in is due to experimental inaccuracy. The overall agreement between experimental and spring-in predictions using the contact boundary condition is however good.

2.0 1.8 1.6 1.4 1.2 1.0 0.8 Spring-in (°) 0.6 0.4 0.2 0.0 40 60 80 100 In-mould cure temp. (°C)

Experiment Free Fix Contact Figure 10. Spring-in after the first cure step, in-mould cure. Error bars represent 95% confidence intervals

Shape distortion at room temperature after the second cure step

During the second cure step the specimens was freestanding, which means that frozen-in deformations are recovered when the specimens pass through the glass transition during heating to the post cure temperature. For this reason the mechanical boundary conditions used during the first cure step (in-mould cure) does not affect the calculated value of spring-in after the second cure step.

22 After the second cure step the predicted spring-in shows good agreement with the experimental values (in-mould cured at 40, 60, 80 respectively 100°C), see Figure 11. The result of 100°C in-mould cure, experimental setup 4, shows the largest difference between predicted and experimental values. Like after the in-mould cure step this was the experiment that showed the largest scatter, specimens 4 and 9 in Table 9. However, the overall agreement between experimental and predicted spring-in is excellent.

2.5

2.0

1.5

1.0 Spring-in (°)

0.5

0.0 40 60 80 100 In-mould cure temp. (°C)

Calculated Experiment Figure 11. Spring-in after the second cure step, experimental values plotted with 95% confidence limits based on pooled standard deviation.

Difference between performing the second cure step freestanding or in the mould.

In Table 11 it was shown that specimens manufactured by experimental setup 5 (in-mould cured during the second cure step) show a significant change in spring-in between the second and third cure step, which the specimens in experimental setup 1-4 (freestanding cured during the second cure step) do not. Previously it was shown that the mechanical boundary conditions using contact to model the interaction between the mould and the specimen is the most accurate description of the interaction between the specimen and the mould during the experiments. For that reason the contact boundary condition was used to simulate spring-in of a specimen manufactured with experimental setup 5.

23 Simulation results of an angle bracket in-mould cured in both step 1and2

Figure 12 shows the predicted change in spring-in during the third cure step for an angle bracket in-mould cured during the two first cure steps. After the second cure step the specimen has reached the final degree of cure with a corresponding glass transition temperature of 136°C and the predicted spring-in is 0.85°. During the third cure step the specimen is heated from 20 to 150°C, the heating starts in glassy state and the spring-in decreases with increased temperature. At the glass transition temperature frozen deformations, due to the constrained first and second in-mould cure steps, are recovered when the polymer softens. This is shown as a vertical jump in Figure 12. Then follows a decrease in spring-in due to heating in rubbery state, which is faster than in glassy state because the through-thickness coefficient of thermal expansion is higher in rubbery state. At 150°C the specimen is kept isothermally but there will not be any isothermal change in spring-in because the specimen was already fully cured after the second cure step. The cooling to the glass transition temperature follows the same path as the heating. Below the glass transition temperature, the spring-in increase with further cooling in glassy state and finally arrive at 1.27°, at room temperature.

1.4

1.2

1.0

0.8

0.6 Spring-in (°)

0.4

0.2

0.0 0 20 40 60 80 100 120 140 160 In-mould cure temp. (°C)

Figure 12. Predicted spring-in during the third cure step for a specimen in- mould cured during the first two cure steps.

24 Comparison of simulation and experimental results

The difference between the spring-in after the second and third cure step is shown in Figure 13. Both from the predictions and experimental results it is clear that the spring-in increase during the third cure step if the specimen is cured in the mould but not if the specimen is free to move during the second cure step. The predictions overestimate the change but the results provide further support for the hypothesis of frozen-ion deformations.

0.32

0.28

0.24

0.20 (°)

 0.16 'T

 0.12

0.08

M 'T 0.04

0.00

-0.04

-0.08

Exp_Free Exp_Const. Calc_Free Calc_Const Figure 13. Change in spring-in during the third cure step, Exp and Calc means experimental respectively calculated values, Free and Const denotes free standing respectively constrained cure during the second cure step.

Conclusions

The simplified material model developed in a companion paper [10] have been validated for a material and cure schedule typical for RTM and autoclave processes. Comparisons between predicted and experimental shape distortion show that the model and simulation approach used capture effects from different cure schedules and after the second freestanding cure step (Figure 11) the overall agreement is excellent. During the in-mould cure three different mechanical boundary conditions were used to model the interaction between the component and the mould. The

25 results showed that the contact boundary condition during in-mould gave closest agreement to the experimental data (Figure 10). Changing the experimental boundary conditions during the second cure step (in-mould cure instead of free standing) demonstrated further the effect from the mould on shape distortion, Table 11. Predicted results showed that the model is able to pick up the responsible mechanisms, even if the predictions overestimated the change in spring-in during the third freestanding cure step, see Figure 13.

Acknowledgement

This work was performed in collaboration with Saab AB and is a part of a national aeronautic research program (NFFP) funded by the Defence Materials Administration in Sweden (FMV). Also the SICOMP Foundation supported this work and the work is also a part of the Integral Vehicle Structure research school (IVS) funded by the Swedish Foundation for Strategic Research.

References

1. Bogetti TA, Gillespie JH. Process-Induced Stress and Deformation in Thick- Section Thermoset Composite Laminate. Journal of Composite Materials 1992;26(5):626-660. 2. White SR, Hahn HT. Process Modeling of Composite Materials: Residual Stress Development during Cure, Part I Model Formulation. Journal of Composite Materials, 1992;26(16):2402-2422. 3. White SR, Hahn HT. Process Modeling of Composite Materials: Residual Stress Development during Cure, Part II Experimental Validation. Journal of Composite Materials, 1992;26(16):2423-2453. 4. White SR, Kim KK. Process-Induced Residual Stress Analysis of AS4/3501- 6 Composite Material. Mechanics of Composite Material and Structures 1998;5:153-186. 5. Prasatya P, McKenna GB, Simon SL. A Viscoelastic Model for Predicting Isotropic Residual Stresses in Thermosetting Materials: Effects of Processing Parameters. Journal of Composite Materials 2001; 35:826-849. 6. Johnston, R. Vaziri, A. Poursartip: Journal of Composite Materials, 2001, 35, 1435-1469. 7. Fernlund G, Rahman N, Courdji R, Bresslauer M, Poursartip A, Willden K, Nelson K. Experimental and numerical study of the effect of cure cycle, tool surface, geometry, and lay-up on the dimensional fidelity of autoclave-

26 processed composite parts. Composites Part A: Applied Science and Manufacturing 2002;33:341-351. 8. Poon H, Koric S, Ahmad MF. Towards a Complete Three Dimensional Cure Simulation of Thermosetting Composites. International Conference on advanced Composites 1998. 9. Huang X, Gillespie JW, Bogetti T. Process induced stress for woven fabric thick section composite structures. Composite Structures 2000;49:303-312. 10.Svanberg JM, Holmberg JA. Predictions of shape distortion, Part I FE- implementation of a path dependent constitutive model. Submitted Composites Part A 11.Svanberg JM, Holmberg JA. An experimental investigation on mechanisms for manufacturing induced shape distortion in homogenous balanced laminates. Composites Part A: Applied Science and Manufacturing 2001;32(6):827-838. 12.Data Sheet: Araldite LY5052/ Hardener HY5052. Ciba Polymers, Switzerland, 1994. 13.Data Sheet: Reinforcements for Composites. Hexcel Fabrics, France, 1997. 14.Shimbo M, Ochi M, Shigeta Y. Shrinkage and Internal Stresses during Curing of Epoxide Resins. Journal of Applied Polymer Science 1981;26:2265-2277. 15.Plepys AR, Farris RJ. Evolution of residual stresses in three-dimensionally constrained epoxy resin. Polymer 1990;31:1932-1936. 16.Nielsen LE. Cross-Linking-Effects on Physical Properties of Polymers. J Macromol Sci –Revs Macromol Chem 1969;C3(1):69-103. 17.Simon SL, McKenna GB, Sindt O. Modeling the Evolution of the Dynamic Mechanical Properties of a Commercial Epoxy During Cure after gelation. J Appl Polym Sci 2000;76(4):495-508. 18.Turi EA, Thermal Characterization of Polymetric Materials, Academic press, 1997 19.Whitney JM, McCullough RL. Micromechanical Material Modelling in Delaware Composites Design Encyclopedia-Volume 2. Technomic Pub. Co. Inc. 1990, 65-72, Technomic Pub. Co. Inc., Lancaster, Pennsylvania, USA 20.Gudmundsson P, Zang W. An Analytic Model for Thermoelastic Properties of Composite Lamiantes Containing Transverse Matrix Cracks. International Journal of Solids and Structures 1993;30:3211-3231. 21.Hull D. An introduction to composite materials, Cambridge: University Press, 1988. 22.Hibbit, Karlsson & Sorensen Inc. ABAQUS/Standard User’s manual, Version 5.8, 1998.

27 Paper D

Shape distortion of non-isothermally cured composite angle bracket

Paper D Shape distortion of non-isothermally cured composite angle bracket

J. Magnus Svanberg

SICOMP AB, Swedish Institute of Composites, Box 271, 941 26 Piteå, Sweden & Luleå University of Technology, Division of Polymer Engineering, 971 87 Luleå, Sweden

ABSTRACT

A simulation tool for prediction of shape distortion and residual stresses has been developed by implementing a new material model in the general purpose FE package ABAQUS. To validate the simulation tool three thick walled angle brackets were manufactured under non-isothermal conditions. Predicted temperatures, degree of cure and shape distortions were then compared with the experimental values for the thick walled angle brackets. Predicted and experimental values show good agreement, which verifies that the simulation procedure is reliable under non-isothermal curing conditions.

1 Introduction

Curing of a composite results in volumetric strains due to thermal expansion and chemical shrinkage. In addition, the mechanical properties change dramatically when the resin transforms from a liquid via a rubber like solid into glassy solid. Furthermore, a significant amount of heat is generated by the cross linking reaction. In other words curing a composite is a quite complex thermal- chemical-mechanical process, which produces challenges in the form of residual stresses and shape distortions. For that reason, there is an interest in use simulation tools to forecast problems due to residual stresses and/or shape distortions. This is not a new problem and researchers have been developing simulation tools for at least ten years, but the industrial use is still limited. To be able to analyse shape distortions and residual stresses a material model is required. The model’s ability to describe the behaviour of the polymer matrix during curing is crucial for the accuracy of the analysis. Because of the complex behaviour of the material during the cure process, a common approximation is to assume that the composite is stress free at the cure temperature and that shape distortions are generated during the cooling from the cure temperature to room temperature1, 2. Using this approximation, only thermal effects are accounted for and chemical shrinkage is neglected. A more accurate approach is to consider residual stress development throughout the entire cure cycle3-8. In an early paper, Bogetti and Gillespie3 presented a one-dimensional model to investigate the effect of gradients in temperature and degree of cure on residual stresses in thick laminates. The model combined cure kinetics, through-thickness heat conduction and elastic laminated plate theory. Johnston et. al.6 developed a plane strain finite element model using a cure-hardening, instantaneous linear elastic constitutive model to predict process-induced deformations of laminated composite structures. The mechanical behaviour of a thermoset polymer during cure is however viscoelastic and rather complicated to model accurately. Prasatya et. al.5 developed an termo-viscoelastic model to predict isotropic residual stresses in three dimensional thermoset materials and showed that an elastic solution overestimated residual stresses and viscoelastic effects can not be neglected. White and Kim developed a rheologically complex visco-elastic FE model4 and investigated curing of a thin rectangular cross ply laminate. They found that the viscoelastic behaviour and the chemical shrinkage of the thermoset composite material have significant effect of the residual stresses. To enable this kind of analysis in a general purpose FE package would be a huge advantage in the analysis of complex composite structures with arbitrary geometries and

2 boundary conditions that might change during the cure process. For example Poon et. al.7 and Huang et. al.8 have used ABAQUS9 for this purpose. The complexity of the viscoelastic material models is, however a problem. Anisotropic viscoelastic materials are difficult and expensive to characterise during cure. In addition FE-simulations using anisotropic viscoelastic material models lead to long calculation times and require much memory for storage of internal state variables. This paper present a simulation tool based on the general purpose FE- package ABAQUS9. The simulation tool uses a simplified stress-strain relationship10 derived from viscoelasticity, which leads to significant savings in computational time, memory requirements and costs for material characterisation as compared to conventional viscoelastic models. The simplified stress-strain relationship has been developed based on observations from a previous work11. This work demonstrates that for a certain range of process conditions (slow processes) the mechanical constitutive relationship for predictions of residual stress and shape distortions do not have to account explicitly for time effects but must account for thermal expansion (different in glassy and rubbery state), chemical shrinkage due to the crosslinking reaction and frozen-in deformations. To validate the simulation tool on a reasonably complicated case, three thick angle brackets were manufactured under non-isothermal conditions. The results from the experiment were then compared with predicted values of temperatures, degree of cure and shape distortion.

Experiment

Three thick 90° angle brackets were manufactured with 300g/cm3 plain E-glass weave (Hexcel 7781)12 and epoxy (LY5052/HY5052)13. Sheets containing 60 layers of the reinforcement were cut to dimensions of 150 x 300 mm and placed symmetrically on a male mould with a radius of 10 mm, see Figure 1. A bag sealed with tacky tape was then placed above the reinforcement and the brackets were manufactured by vacuum infusion under well controlled thermal conditions. The initial temperature of the mould was about 30°C, the heating system of the mould was turned on at the beginning of the injection, and the mould was heated to 35°C, which was maintained during the injection. After injection and prior to gelation, the mould was heated to 80°C and kept there for 2 hours.

3 Figure 1. Injection of angle bracket.

After the first cure step, the angle brackets were sliced into 30 mm wide specimens. One specimen from each of the three manufactured angle brackets was then cured, free standing, in an oven at 150°C for 2 hours in a second cure step. During the first cure step the temperature was measured with thermocouples at three different locations; between the laminate and tool, in the middle of the laminate and between the laminate and the bag. In addition to the temperatures, the glass transition temperature (Tg) and spring-in were also measured after the first and second cure step. The spring-in is mainly affected by the conditions in the bent region. Hence the thickness was measured in this region to 12.5 mm (standard deviation 0.7 mm) based on three 30 mm wide specimens. A systematic reduction in thickness of about 0.7 mm from inlet to outlet is also present, but that will not be considered any further in this paper.

Simulation

A coupled thermal-mechanical analysis is required to be able to simulate the experiment presented in this paper, at least for the first cure step. When the temperatures at the boundaries of the bracket are known from the experiment they can be input to the simulation and it can therefore be assumed that shape

4 distortions during the curing step do not affect the thermal boundary conditions. Stresses and deformations then depend on temperature and the degree of cure, but temperature and degree of cure can be found without knowledge of stresses and deformations. A sequential coupled analysis is therefore suitable in this case. The analysis is performed by first conducting an uncoupled heat transfer analysis. The results, which in this case are temperatures and degrees of cure, are stored on files as functions of time. The following stress/deformation analysis is then executed and the results from the thermal analysis are used as input to calculate residual stresses and shape distortion. The alternative to a sequential analysis is a fully coupled thermal-mechanical analysis where the thermal and stress/deformation equations are coupled both ways. This kind of analysis would have been required if the temperatures at the boundaries were unknown and shape distortion during the cure process affects the thermal boundary conditions. A fully coupled analysis of this kind requires accurate thermal and mechanical modelling of the interaction between the tool and the component. This is outside the scope of the present paper as well as thermal modelling of tool, bag and the surrounding air, which would have been necessary if the temperatures at the boundaries not had been known. Constitutive models

During the cure process latent heat is released in the resin. The energy release rate per unit volume, q is proportional to the reaction rate and is expressed as, χ χρ=⋅ ⋅d ⋅− qT(,)rtot H (1 V f ) (1) dt ρ χ where r , H tot ,Vf and ddtare density of the resin, total heat released during the curing reaction (energy per unit mass), fraction fibre by volume, and chemical reaction rate, respectively. The latent heat is entered into the equation for heat conduction as,

∂∂∂ ∂∂ ∂∂  ρλλλχ⋅⋅TT = + T + T + cqTxyz (,) (2) ∂∂tx ∂ x ∂ y ∂ y ∂ z  ∂ z where ρ is the density of the laminate, c is specific heat capacity T is λ = temperature and i ,(ixyz , , ) are the heat conductivity in the x, y and z direction. It was very convenient to implement the heat generation into ABAQUS using the user subroutine HETVAL, others have done this in a similar way7, 8.

5 Em

Ea ηm

Figure 2. Standard linear solid (SLS).

The basic idea of the mechanical constitutive relationship can be explained by considering a linear isotropic viscoelastic solid14. To keep the presentation simple, a standard linear solid is considered in which a spring is placed in parallel with a Maxwell element, see Figure 2. The stress-strain relationship of such a material is,

t ∂ε σττ=−Et() d ∫ ∂τ (3a) 0 where ε is strain and the relaxation modulus E(t) is given by,

E −⋅m ()t η =+⋅m Et() Eam E e (3b) The relationship is then simplified by assuming that the relaxation is instantaneous at temperatures above the glass transition, in the rubbery state, by letting the viscosity, ηm, go to zero. The stiffness in the rubbery state, Er is then equal to Ea. At temperatures below the glass transition temperature it is assumed that there is no relaxation, by letting the viscosity ηm,gotoinfinityandthe stiffness in glassy state, Eg is then Ea +Em. Eqn. (3) is then rewritten as,

6  t ∂ε EEEdTT⋅+ετη∫() −,when < , →∞ σ =  rgr∂τ gm t1 (4)  ⋅≥→εη  ETTrg,,when,0m where t1 is the time when the last rubbery-glassy transition took place. The rubbery stiffness, Er and the glassy stiffness Eg can be found by mechanical testing. A simplified stress-strain relationship without an explicit time effect has been achieved. A detailed description of the stress-strain relationship is presented in Ref. (10). < When TTg Eqn. (4) is written in three dimensions as,

t ∂ε σε=⋅+CCCdrgr() − j τ iijj∫ ijij∂τ (5) t1 ≥ When T Tg this simplifies to,

σε=⋅r iijjC (6)

σ ε r where i and j are stress and strain in Voigt notation. The stiffness matrices Cij g and Cij can be identified as rubbery and glassy stiffness, respectively. The free expansion increment is,

∆ε free = ∆ε T + ∆ε C χ i i (T ) i ( ) (7) and consists one thermal and one chemical part denoted by superscript T and C. The thermal part is expressed as

∆ε T = α ⋅ ∆ i i T (8) where the coefficient of thermal expansion, α is different in the glassy and rubbery state,

αχg ,TT< () α =  ig i αχr ≥ () (9)  ig,TT

The chemical shrinkage is described in similar way as,

∆ε C = β ⋅ ∆χ i i (10) but without dependency on the material state for the coefficient of chemical shrinkage, β. The user subroutines UMAT and UEXPAN were used to implement the mechanical constitutive relationship respectively the expansion feature into

7 ABAQUS.UMAThavebeencommonlyusedbyotherstoimplementdifferent types of constitutive models7, 8. The FE-implementation together with a detailed description of the material model is presented elsewhere10. FE-model

As previous mentioned only the angle bracket was modelled, both the tool and the bag were left out since the thermal history at the boundaries is known from experiment. The angle bracket was modelled (see Figure 3) in ABAQUS9 using diffusive heat transfer 8-node linear brick elements (DC3D8) in the thermal analysis and stress/displacement 8-node linear brick elements (C3D8) in the subsequent stress/deformation analysis. Only the first cure step was thermally analysed. The nodal temperatures on surfaces in contact with the bag or the tool were set to follow the temperatures known from the experiments.

Figure 3. Distorted angle bracket.

Two types of mechanical boundary conditions were investigated for the first cure step, free and fully constrained. When the angle bracket is modelled as freestanding, shape distortions develop during the entire first cure step (in- mould cure). If the bracket instead is modelled as fully constrained to the rigid male mould half during the first cure step residual stresses are formed instead of shape distortions. In this case the shape distortion is formed at demoulding when the residual stresses are relaxed. In reality, a boundary condition between these

8 two extremes can be expected. The second cure step (the postcure) was modelled as freestanding during the entire step and the temperature considered as homogenous in the material.

Material properties used in the simulation

The specific heat (cp) in the glassy state of the laminate (angle bracket) shown in Figure 4 is for fully cured specimens as determined by DSC measurements according ASTM E1269. The possibility to using a variable cp is not implemented yet and therefore a constant value of 1.1 [J/g °K] was used in the following calculations, which corresponds to the value at the in-mould cure temperature.

1.15

1.10

1.05 (J/g/°K) p C

1.00

0.95 40 45 50 55 60 65 70 75 80 85 90 Temp. (°C)

Figure 4. Specific heat of fully cured neat resin.

The thermal conductivity, λ [W/m °K] of fully cured neat resin was measured to 0.3 by a device developed at Chalmers University of Technology in Sweden called Hot Disc Thermal Constants Analyser. Thermal conductivity of fibreglass15 is 1.0 [W/m °K]. The unidirectional ply values were calculated 16 using micro-mechanic and the thermal conductivity for the 0/90s lay-up is estimated17 as, λλ== λλ + = + = XY( LT )/2 (0.70 0.57)/2 0.63W/m°K (11a)

λλ== ZT0.57 W/m °K (11b)

9 The reaction rate in eqn. (1) is described by eqn. (12)18. The parameters in eqn. (12) were found by fitting the model parameters to data determined by isothermal DSC measurements at different temperatures19.

d χ −52600 =⋅⋅RT χχ0.17 − χ 1.83 110000e (max ) (12a) dt

χ =+⋅ max 0.782 0.002 T (12b) where χ,Rand T are degree of cure, the gas constant and cure temperature, respectively. The mechanical properties for the resin LY5052/HY5052 in glassy and rubbery state presented in Table 1 were found in Refs. (19-21).

Table 1. Mechanical properties of neat LY5052/HY5052. Glassy state Rubbery state Property Symbol Source Value Symbol Source Value g r r ⋅ ⋅ + ⋅ν r Young’s modulus (GPa) Em 19 2.6 Em Gm 2 (1 m ) 0.0028 ⋅ r − ⋅ r ν g ν r 3 Km 2 Gm m m ⋅ r + ⋅ r Poisson’s ratio (---) 19 0.38 6 Km 2 Gm 0.497 -6 α g α r ⋅α g a. Coeff. of therm. exp.(10 /°C) m 13 71 m 2.5 m 178 E g g g m r Gm b. Gm ⋅ +⋅ν g Gm Shear modulus (GPa) 2 (1 m ) 0.94 100 0.0094 g g α 20 g Em r m ⋅ g K m ⋅ − ⋅ν g K m r Km Bulk modulus (GPa) 3 (1 2 m ) 3.6 α 1.4 m a. Assumed value based on Ref. (21). b. Assumed

The volumetric chemical shrinkage of the resin was determined to –7 %. It may be argued that this is too much for an epoxy. However, the estimate is based on density measurements at 30°C of isothermally cured neat resin at the same temperature. Because the resin was both, cured and measured at the same temperature the density change is only a result of chemical shrinkage. The density measurements showed a volumetric shrinkage of –5 %. A DSC measurement showed that the resin only reach a degree of cure of 71% at this cure schedule, which must also be taken in consideration. This was done by assuming a linear relationship between volumetric chemical shrinkage and degree of cure, which leads to total volumetric chemical shrinkage of –7%.

10 Mechanical properties of fibreglass was found in Ref. (15); Young’s modulus, Poisson’s ratio and coefficient of thermal expansion are 76 GPa, 0.22 and 4.9⋅10-6 1/°C respectively. The average fibre content in the manufactured angle brackets was 56% by volume, based on three samples (with fibre contents of 56.6, 57.3 and 54.5%) measured by pyrolysis (560°C for 4h). Based of the fibre fraction, the lay-up

(0/90)60 and the mechanical properties of fibre and matrix the mechanical properties of the laminate (both in the glassy and the rubbery state) was calculated using a micro-mechanic model16 and three dimensional laminate theory22,seeTable2.

Table 2. Mechanical properties of glass fibre reinforced LY5052 / 7781

(Vf=56%). Symbola Glassy state Rubbery state

Ex (GPa) 26.2 21.4 Ey (GPa) 26.2 21.4

Ez (GPa) 10.1 2.7 νxy (-) 0.094 0.0021 νxz (-) 0.46 0.83 νyz (-) 0.46 0.83

Gxy (GPa) 3.0 0.033 Gxz (GPa) 2.9 0.033

Gyz (GPa) 2.9 0.033 -6 αx (10 /°C) 13.9 5.4 -6 αy (10 /°C) 13.9 5.4 -6 αz (10 /°C) 58.7 229.0 -3 -5 βx (-) -3.2⋅10 -7.4⋅10 -3 -5 βy (-) -3.2⋅10 -7.4⋅10 -2 -2 βz (-) -1.9⋅10 -3.2⋅10 a. E, ν,G,α and β are Young’s modulus, Poisson ratio, Shear modulus, Coefficient of thermal expansion and Coefficient of chemical shrinkage respectively.

Results and Discussion

The temperatures at the laminate and tool, in the middle of the laminate and the laminate bag interface were measured every other second during injection and curing of the three angle brackets. The temperature readings from three angle brackets were averaged at each location and time relative to the start of the

11 injection. The averaged temperature at each location versus time is presented in Figure 5.

100

90

80

70

60

50

40 Temp. (°C) 30

20

10 Bag Middle Tool 0 0 20 40 60 80 100 120 time (min.)

Figure 5. Average temperature in the laminate at three different positions.

The highest temperature (94°C after 59 minutes) was reached in the middle of the laminate. The glass transition temperature (Tg) and spring-in were also 23 measured after the first and second cure step. Tg was measured using DSC (40°C/min heating rate) at approximately the same location, as the temperatures were measured during the first cure step. From Tg it is then possible to get an estimate of the degree of cure, χ in the laminate by using eqn. (13)19. =−⋅−χ D TCg 128 250 (1 ) (13) Predicted and measured degrees of cure at the end of the first cure step are presented in Table 3. From Table 3 is it obvious that the prediction of degree of cure is under estimated, but the deviation is particularly large considering that the cure kinetics and thermal properties of the composite were determined by micromechanics and properties of the constituents.

12 Table 3. Degree of cure after the first cure step. Degree of cure (-) a b Position Tg (qC) Measured Predicted Bag 90 0.85 0.82 Middle 101 0.89 0.86 Tool 107 0.92 0.87 a. Measured by DSC.

b. Calculated from Tg with eqn. 13

In the simulation, both the tool and the bag were omitted and only the angle bracket was modelled. The nodal temperature on the surfaces in contact with the bag and the tool were set to follow the temperatures known from the experiments. This means that the thermal boundary conditions used in the simulation should be reasonable accurate, which was confirmed by the good agreement, shown in Figure 6, between predicted and measured temperature in the middle of the laminate.

100

90

80

70

60

Temp. (°C) 50

40

30 Measured Predicted 20 0 20406080100120 Time (min.)

Figure 6. Temperature in the middle of the laminate.

Shape distortion in form of an angle change of an angle bracket is denoted spring-in, see Figure 3. The spring-in angle was measured at room temperature after the first and second cure step. The results are presented in Table 4 and are averages based on three angle brackets. An angle gauge was used and the measurement was performed by hand, the resolution of the measurement is roughly ± 0.1°. Unfortunately, two of the angle brackets were affected by fibre wrinkling in

13 the radius during manufacturing but this problem did not seem to affect the spring- in. Table 4. Comparison between predicted and measured spring-in, standard deviation in parenthesis. Cure step 1 2 Boundary Free Constr. Free condition Predicted 1.4° 0.5° 1.2° Measured 1.0° (0.10°)1.1° (0.05°)

Figure 7 shows predicted spring-in versus curing time during the first cure step (in-mould cure). Results presented both free and fully constrained boundary conditions. During freestanding cure, the specimen is not constrained by the mould and the spring-in starts to develop at point Afree in the figure, which correspond to gelation of the resin. The knee between point Afree and Bfree is glass transition (vitrification) where the resin transforms from the rubbery state to the glassy state.

1.6 1.4 Cfree Bfree 1.2

1.0 0.8 0.6 Cconst. Spring-in (°) 0.4 0.2 Afree 0.0 -0.2 0 50 100 150 200 time (min)

Constrained Free

Figure 7. Cure step 1, predicted spring-in

Between point Bfree and Cfree thespecimeniscooledtoroomtemperature. This cooling was simulated as an instantaneous and uniform temperature jump, which is a reasonable assumption because the part is already in the glassy state, the time dependent chemical reaction is fairly slow and chemical shrinkage can be neglected during the cooling. In the case of a fully constrained specimen, no

14 shape distortions are formed prior to demoulding. This demoulding corresponds to the vertical thick line in Figure 7. The shape distortion for this case is smaller than for the freestanding case owing to deformations that are frozen into the material; Ref. (10) gives an in depth explanation of the mechanisms involved during the cure process. During the second cure step, only a small amount of the chemical reaction is left and hence only a small amount of exothermal energy will be released during the second cure step. This means that the temperature during the rest of the analysis can be assumed homogeneous in the material, or in other words spatially uniform. Regarding the degree of cure it was previously shown in Table 3 that the degree of cure is not uniform after the first cure step. Therefore, at the start of the second cure step the degree of cure is not homogenous but becomes homogenous after the cure step.

Point Cfree in Figure 8 is the same point as in Figure 7 that is the freestanding specimen after completed in-mould cure and cool-down but before heating for post cure. The knee at point D in Figure 8 is a result of softening of the polymer at the glass-rubbery transition during heating. The knee is not sharp because the degree of cure is not spatially uniform at this moment owing to the non- isothermal, in-mould cure, which in turn means non-uniform glass transition temperatures in the material. Further, at point E additional chemical shrinkage develop owing to additional chemical reaction during the second cure step. Point F is the rubbery-glass transition during cooling. The transition is at a higher temperature than during heating (point D)becausethedegreeofcurehas increased and hence the glass transition temperature. Note also that the knee at point F is much sharper than at point D because the degree of cure is now spatially uniform in the material, which was not the case during heating. Finally point G is predicted spring-in at room temperature after complete cure. Figure 9 shows the same scenario as Figure 8 but for a specimen fully constrained during the first cure step. The difference between Figure 8 and Figure 9 is that constrained in-mould cure results in deformations that are frozen into the specimen10. These are released when the specimen for the first time is heated freestanding above its range of glass transition temperatures, point D. The shape of the curve at point D is attributed to non-uniformity in the material, and therefore variations the glass transition temperature, so that the polymer softens at different temperatures during heating.

15 1.6 Cfree 1.4 G 1.2 D 1.0 F 0.8

0.6 Spring-in (°) 0.4

0.2 E 0.0 0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00 160.00 -0.2 Temp. (°C)

Figure 8. Predicted spring-in during cure step 2, free standing cure during cure step 1.

1.6

1.4 G 1.2

1.0 F 0.8 Cconst. 0.6 Spring-in (°) 0.4 D 0.2 E 0.0 0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00 160.00 -0.2 Temp. (°C)

Figure 9. Predicted spring-in during cure step 2, fully constrained cure during cure step 1.

In Table 4 measured spring-in after the first and second cure step is compared with predicted values. After the first cure step, the predictions using free boundary conditions over estimate the shape distortion. The opposite occurs

16 when fully constrained boundary conditions are used, which under estimate the shape distortion. This is encouraging since in the real case the part is to some extent constrained but not fully constrained by the tool. Predicted and measured shape distortion after the second cure step is in very close agreement (1.2° predicted compared to the measured 1.1°).

Conclusions

This paper has shown that the simulation slightly under estimates the degree of cure in the angle bracket. The difference is however acceptable, in particular when considering the fact that the cure kinetic model, eqn. (12) and glass transition function, eqn. (13) was developed for neat resin. The temperature in the middle of the laminate agrees well with measured temperatures and is a result of that the surface of the bracket was prescribed to follow measured values. This shows that accurate thermal boundary conditions should give accurate simulation results. The measured values of spring-in agree with the predicted results at least after the second cure step. Spring-in after the first cure step was predicted using two different boundary conditions, free or fully constrained and it is shown that the different mechanical boundary conditions results in different spring-in predictions. In reality a boundary condition between these two extremes is expected, which also is what the results in Table 4 suggest.

Acknowledgement

The author gratefully acknowledges the efforts of Magnus Edin at SICOMP AB for manufacturing angle brackets. This work was performed in collaboration with Saab AB and is a part of a national aeronautic research program (NFFP) funded by the Defence Materials Administration in Sweden (FMV). The Swedish Foundation for Strategic Research also supported this work through the Integral Vehicle Structure research school (IVS).

References

1. W.J. Jun, C.S. Hong: Journal of Reinforced Plastics Composites, 1992, 11, 1352-1366. 2. S.C. Tseng, T.A. Osswald: Journal of Reinforced Plastics Composites, 1994, 13, 698-721. 3. T.A. Bogetti, J.W. Gillespie: Journal of Composite Materials, 1992, 26, 626- 660

17 4. S.R. White, Y.K. Kim: Mechanics of Composite Material and Structures, 1998, 5, 153-186. 5. P. Prasatya, G.B. McKenna, S.L. Simon: Journal of Composite Materials, 2001, 35, 826-849. 6. Johnston, R. Vaziri, A. Poursartip: Journal of Composite Materials,2001,35, 1435-1469. 7. H. Poon, S. Koric, M.F. Ahmand: Proc. Int. Conf. on ‘Advanced Composites’, Hurghada, Egypt, December 1998, 165-180, Editors: Y. Gowayed, Department of Textile Engineering, Auburn University, Alabama, USA and F. Adb El Hady, Department of Automotive Engineering, Ain Shams University, Cairo, Egypt 8. X. Huang, J.W. Gillespie, T.A. Bogetti: Composite Structures, 2000, 49, 303- 312. 9. Hibbit, Karlsson & Sorensen Inc: ‘ABAQUS/Standard User’s manual’, Version 5.8, 1998. 10.J.M. Svanberg, J.A. Holmberg: ‘Predictions of shape distortion, Part I FE- implementation of a path dependent constitutive model’, Submitted to Composites Part A 11.J.M. Svanberg, J.A. Holmberg: Composites Part A, 2001, 32, 827-838. 12.Data Sheet: ‘Reinforcements for Composites’, Hexcel Fabrics, France, 1997. 13.Data Sheet: ‘Araldite LY5052/HY5052’, Ciba Polymers, Switzerland, 1994. 14.R.M. Christensen: ‘Theory of viscoelasticity’, 2nd edn, 1982, New York, Academic Press. 15.D. Hull: ‘An introduction to composite materials’, 1981, Cambridge, University Press. 16.J.M. Whitney, R.L. McCullough, ‘Micromechanical Material Modelling, Delaware Composites Design Encyclopedia-Volume 2’, 1990, 65-72, Technomic Pub. Co. Inc., Lancaster, Pennsylvania, USA 17.W.C. Tucker, R. Brown: Journal of Composite Material,1989,23, 787-797. 18.J.M. Kenny, A. Maffezzoli, L. Nicolais: Composite Science and Technology, 1990, 38, 339-358. 19.J.M. Svanberg, J.A. Holmberg: ‘Results from material and spring-in characterisation’, SICOMP Technical Note 99-006. SICOMP AB, Box 271, SE-941 26 Piteå, Sweden, 1999. 20.A.R. Plepys, R.J. Farris: Polymer, 1990, 31, 1932-1936. 21.M. Shimbo, M. Ochi, Y. Shigeta: Journal of Applied Polymer Science, 1981, 26 2265-2277. 22.P. Gudmundsson, W. Zang: International Journal of Solids and Structures, 1993, 30, 3211-3231.

18 23.E.A. Turi: ‘Thermal Characterization of Polymetric Materials’, 2nd edn, 1997, New York, Academic press.

19 Paper E

Prediction of shape distortions for a curved composite C- spar

Paper E Prediction of shape distortions for a curved composite C- spar

J. Magnus Svanberg a, b, Christina Altkvistc and Tonny Nymanc

aSICOMP AB, Swedish Institute of Composites, Box 271, 941 26 Piteå, Sweden bLuleå University of Technology, Division of Polymer Engineering,971 87 Luleå, Sweden cSaab AB, SE-581 88 Linköping, Sweden

Abstract

During manufacturing of a structural component made of composite material, shrinkage and shape distortions occurring. To achieve a component with desired shape the mould has to be compensated for these shape distortions. Today, compensation of the mould is a time and cost consuming task that requires a lot of experience. The possibility of performing this task with assistance of a simulation tool is considered as an important step towards a more effective manufacturing of complex shaped components. This paper presents predictions and validations of shape distortions of a curved C-spar, using a simulation tool developed in the general purpose FE- program ABAQUS. The simulation tool is based on a simplified mechanical constitutive model that accounts for the mechanisms identified in a previous experimental study concerning the influence from the cure schedule on shape distortions. The main objective of the present paper is to validate the simplified constitutive model and a simplified simulation approach. The feasibility for simulation of shape distortions of a component with relatively complex shape has also been investigated. The results show, for the material and manufacturing process considered in this work, that the simulations give a good estimate of cure induced shape distortions, even if a number of assumptions and simplifications have been used. The work also shows that the process model and simulation approach used makes it possible to predict shape distortions of fairly large and complex shaped parts with reasonable labour and computer resources.

1 Introduction

A well-known manufacturing process for high performance composites is RTM (Resin Transfer Moulding) where dry reinforcement is placed in a mould cavity and the resin is injected into the cavity. After the reinforcement has been impregnated, the material is kept in the mould for a sufficient time to allow the thermoset to cure. During this in-mould cure there are two transformations of the resin, first a transformation from a liquid to a solid, the gelation [1]. At this point the glass transition temperature of the resin is lower than the cure temperature and the resin is in the rubbery state. When the cure process proceeds the glass transition temperature increases and eventually exceed the cure temperature. This second transformation is called the vitrification [1]. At this point the resin transforms from the rubbery state to the glassy state, with a dramatic change in mechanical properties. It is usually after this point the mould is opened and the component is released, the so-called demoulding. In Figure 1 the manufacturing steps in RTM are illustrated.

Mould Resin Air Air Fibres

Preforming Injection Curing Demoulding

Figure 1. Illustration of RTM.

The resins used in high performance composites usually require elevated cure temperatures to start and maintain the cure reaction. Consequently, the component shrinks when it is cooled to room temperature after completed cure. During the cure reaction, linear polymer chains are cross-linked to a denser three-dimensional molecular structure and therefore a composite component also shrinks due to the chemical shrinkage of the polymer during cure. Because of the anisotropic nature of the material, shrinkage does not only mean that the dimensions of a component are scaled down, the shape will also change. Even if the laminate is through-thickness homogeneous and isothermally cured, shrinkage leads to an angular decrease of an angle section called spring-in, ∆θ [2], see Figure 2.

2 ∆θ

Figure 2. Spring-in

During cure a considerable amount of reaction energy is released from the polymer, which may cause gradients in temperature and degree of cure leading to shape distortions. In general, this is not a problem for a thin walled structure but it might be a problem for a thick walled structure in combination with poor heat transfer properties for the mould. The gradients in temperature, degree of cure and shrinkage are not the only contribution to shape distortions, other parameters are for instance unsymmetrical lay-up and gradients in fibre fraction [2]. There are a number of challenges to design a functioning mould. One of them is compensation for cure induced shape distortions. When a mould is designed for a component with high shape tolerances, a lot of effort needs to be spent on compensation of the mould geometry in order to obtain a component with desired geometry. Usually there is no problem to compensate a single curved geometry, such as an angle bracket, based on experience and rules of thumb. But, when the component geometry becomes more complex the compensation of the mould for shape distortions is much more difficult and sometimes it is necessary with a number of modification cycles or even a complete redesign of the mould before the right shape of the component is achieved. There are a lot to save, in both development time and costs, if the number of modifications and test runs can be reduced. The possibility of performing this task with assistance of a simulation tool is considered as an important step towards a more effective manufacturing of complex shaped components. The cure process is a fairly complex procedure including thermal, chemical as well as mechanical phenomena. Accordingly, a comprehensive simulation tool for residual stress and shape distortions has to contain heat conduction, cure kinetics and a suitable mechanical constitutive model. In the beginning of the nineties both Bogetti and Gillespie [3] and White and Hahn [4, 5] presented simulation tools for residual stresses and shape distortions. Bogetti and Gillespie

3 [3] used an elastic laminated plate theory in combination with heat conduction and cure kinetics and analysed thick section thermoset composite laminates. White and Hahn [4, 5] used a viscoelastic material model in combination with cure kinetics and analysed thin, unsymmetrical cross-ply carbon/bismaleimid laminates. Later White and Kim [6] developed a thermo-chemo-rheologically complex viscoelastic finite element model and investigated curing of a thin rectangular cross ply laminate. Prasatya et al. [7] predicted isotropic residual stresses in three dimensionally constrained neat thermoset material using a viscoelastic model. The material was cured isothermally, which implies that it was possible to capture the degree of cure phenomena by only a cure kinetic simulation without solving the heat conduction equation. Various types of linear elastic constitutive models have been used. For instance a plane strain finite element model using a cure-hardening, instantaneous linear elastic (incremental linear elastic) constitutive model to predict process-induced deformations of laminated composite structures was developed by Johnston et al. [8]. Fernlund et al. [9] recently used this model to investigate effects of cure cycles, tool surfaces, geometry and lay-up on shape distortion. A general purpose FE-program like ABAQUS [10] include both heat conduction and mechanical constitutive models and there are possibilities to implement various user defined models as subroutines. This makes it possible to simulate residual stresses and shape distortions. One advantage of using such a program is the ability to handle complex geometries. Another advantage is that these types of programs usually have well developed functionality for pre- and post processing. For example Poon et al. [11] and Huang et al. [12] used ABAQUS for this type of analysis. Poon et al. [11] implemented cure kinetics and a viscoelastic temperature and cure dependent material model and analysed curing of a bent plate and a hollow cylinder. Huang et al. [12] used a cure dependent textile unit cell model in combination with ABAQUS to predict process induced stresses for woven fabric composite structures. This paper presents predictions and validations of shape distortions of a curved C-spar. Predictions of shape distortions were performed using a simulation tool developed within ABAQUS. The simulation tool is based on a simplified mechanical constitutive model [13] and accounts for the mechanisms identified in a previous experimental study regarding the effect from the cure schedule on shape distortions [14]. The model has previously been validated for prediction of shape distortions of thin isothermally cured fibreglass/epoxy single curved composites with good results [15]. The mechanical constitutive model has also successfully been used in combination with heat conduction and cure

4 kinetics to predict shape distortions of a non-isothermally cured thick single curved geometry [16]. The main objective of the present paper is to validate the simplified constitutive model and a simplified simulation approach for a different material system and geometry and also to investigate the feasibility for simulation of shape distortions for a complex shaped component.

Component definition

The demonstrator component chosen for validation of the constitutive model and the simulation approach have been developed within the European research project PRECIMOULD and the results are used with permission from BAESYSTEMS on the behalf of the PRECIMOULD consortium. The CAD description of the C-spar was delivered by BAESYSTEMS in terms of a CATIA model, Figure 3. The spar contains two integral cleats. One end of the spar is left open and the other is closed.

Figure 3. C-spar. Courtesy of BAESYSTEMS on behalf of the PRECIMOULD consortium.

The dimensions are: • Length (centre arc): 800 mm • Width (web): 100 mm • Depth (flanges): 46 mm • Sweep angle: 27°

5 Three C-spars, with a fibre volume fraction of 55%, were manufactured by the Resin Transfer Moulding (RTM) technique using a steel mould. The mould was designed using the corrected shape data produced from the FE-based simulation capability developed within the PRECIMOULD project. Manufacturing of the tool was carried out by BAESYSTEMS. The material used for manufacturing of the components was a balanced carbon fibre 2x2 twill weave, Hexcel G986 and the epoxy resin RTM6 [17]. The lay-up was [+45/0/-45/90/+45/0]s12, with a total nominal thickness of 3.48 mm. The manufacturing can be summarised as follows:

1. The preform was placed in the mould and the mould was heated to 120°C. 2. Resin was injected at 80°C into the mould at 120°C. Injection pressure was 0.1-0.4 MPa. 3. Outlet valve was closed and pressure maintained for 30 minutes during which the temperature of the resin was increased to 120°C. 4. Inlet valve was closed and the temperature increased to 180°C. 5. Curing for 150 minutes at 180 +10/-5 °C. 6. Mould temperature was reduced to room temperature before the component was removed from the mould.

The manufactured C-spars were dimensionally checked using a Coordinate Measure Machine (CMM) and the actual component spring-in was calculated in the positions shown in Figure 4. Inner 1 8 2 7 6 345

1 8 2 7 3 6 4 5

Outer Figure 4. Definition of cross-sections. Courtesy of BAESYSTEMS on behalf of the PRECIMOULD consortium.

6 Material model

The component in the present work is fairly thin and manufactured with controlled mould temperature. This indicates that no significant exothermal temperature peaks occur during cure. The temperature is approximately isothermal and both degree of cure and temperature are spatially uniform, which implies that the same simulation approach as in [15, 18] can be used. The chemical shrinkage and thermal contraction are, in the present case, two separate events i.e. the evolution of degree of cure takes place at isothermal conditions at the cure temperature and the cooling after complete cure occurs without further evolution of degree of cure. In this section a brief presentation of the material model used for shape distortion predictions is made. A detailed description of the material model is given in [13]. Mechanical constitutive model

The basic idea of the mechanical constitutive relationship can be explained by considering a linear viscoelastic solid [19]. To keep the presentation simple, a standard linear solid is considered in which a spring is placed in parallel with a Maxwell element. The one-dimensional stress-strain relationship of such a material is,

t ∂−()εεE στ=−Et() d τ (1a) ∫ ∂τ 0 where ε is total strain and ε E is expansional strain, described in the next sub section. The relaxation modulus E(t) is given by,

−E m ⋅t η =+⋅m Et() Eam E e (1b) The relationship is then simplified by assuming that the relaxation is instantaneous at temperatures above the glass transition, in the rubbery state, which correspond to a viscosity, ηm close to zero. The stiffness in the rubbery state, Er, is then equal to Ea. At temperatures below the glass transition temperature it is assumed that there is no relaxation, which correspond to an infinite viscosity ηm, and the stiffness in glassy state, Eg, is then identified as Ea

+Em. The rubbery stiffness, Er and the glassy stiffness Eg can be found by mechanical testing. A simplified stress-strain relationship without an explicit time effect has been achieved and Eqn. (1) is rewritten as,

7  ⋅−εεE ≥ η → ETTXr (),when(),0g m  σ =  t ∂−()εεE (2) EEEdTTX⋅−()εεE +∫() − τ,when(), < η →∞  rgr∂τ gm  t1 where t1 is the time for the last rubber-glass transition. The glass transition temperature, Tg depends on the degree of cure, X,as, = TfXg () (3) In the literature various relations between glass transition temperature and degree of cure have been used. For example, Holmberg [23] used a linear relationship, others have used the DiBenedetto equation [20], e.g. Simon et al. [21]. In three dimensions Eqn. (2) is written as,

 rE⋅−εε ≥ η → CTTXijkl() kl kl,when(),0 g m  σ = t E ij  ∂−()εε (4) rEgr⋅−+εε −kl kl τ < η →∞ CCCdTTXijkl()() kl kl∫ ijkl ijkl,when g (),  ∂τ m  t1 σ ε r where ij and kl are the stress and strain tensor. The stiffness tensor Cijkl and g Cijkl can be identified as rubbery and glassy stiffness, respectively.

Expansional strains The expansional strain in Eqns. (1-4) is decomposed in a thermal and a chemical part as,

εεεETC=+ ij ij ij (5) where superscript T and C denote thermal and chemical strain. The thermal part is given by,

t ∂T εαT = (,TX ) dt′ (6a) ij∫ ij ∂ ′ 0 t where the instantaneous coefficients of thermal expansion αij depend on temperature and degree of cure as,

α l ,andXX<≥ TTX()  ij gel g αα=≥r ≥() ij ij,andXX gel TTX g (6b) α g < ()  ij, TTX g

8 α l α r α g ij , ij and ij are linear coefficient of thermal expansion in the liquid, rubbery and glassy state, respectively. Tg is the degree of cure dependent glass transition temperature and Xgel denote degree of cure at gelation. The free chemical shrinkage of neat resinisassumedtobeproportionalto the degree of cure. However, during free chemical shrinkage of a composite the fibres will impose a constraint on the resin. For the composite this leads to different chemical shrinkage in the rubbery and glassy state since the rubbery and glassy modulus of the resin is significantly different. In accordance with this we define the chemical shrinkage as,

t ∂X εβC = (,TX ) dt′ (7a) ij∫ ij ∂ ′ 0 t where the instantaneous coefficients of chemical shrinkage βij depend on temperature and degree of cure as,

β l ,andXX<≥ TTX()  ij gel g ββ=≥r ≥() ij ij,andXX gel TTX g (7b) β g < ()  ij,TTX g

β l β r β g where i , i and i are the linear coefficients of chemical shrinkage in the liquid, rubbery and glassy state. The constitutive model was written on incremental form following the procedure of Zocher et al. [22] and implemented in ABAQUS using the user subroutine UMAT. The user subroutine UEXPAN was used to implement expansional strain into ABAQUS using a forward Euler scheme. A detailed description of the implementation is presented in [13].

FE-analysis

FE-model

The model consists of approximately 30 000 nodes and 4500 element with one element in the thickness direction. The model is built up using 15-node quadratic triangular prism elements (C3D15 in ABAQUS notation) and 20-node quadratic brick elements (C3D20) in ABAQUS VERSION 5.8-10 [10]. Five elements were used around the corner radii. Modelling and analysing an open cross-section of the C-spar using two different mesh densities verified the accuracy of the mesh. The first mesh was equal to the C-spar in Figure 5 and the other was modelled with five elements in

9 the thickness direction and 16 elements around the corner radii. The results showed that the difference between the two analyses was about 1% and the mesh in Figure 5 is considered as suitable for further analysis.

Figure 5. FE-model of the C-spar.

Material properties

The mechanical properties of the RTM6 resin [17] both in glassy and rubbery state are presented in Table 1 and Table 2, respectively. The gelpoint for this particular resin occurs at a degree of cure of approximately 60% [23] and the expression used to relate glass transition temperature to degree of cure is given by [23], =−⋅− TXg 230 467 (1 ) (8) The glassy and rubbery mechanical properties of the laminates are presented in Table 4. The mechanical properties of the laminate were estimated from the properties of the constituent materials by self-consistent field micro mechanics [24] and three-dimensional laminate theory [25]. A knock down factor to reduce the longitudinal Young’s modulus of the fibres was used to compensate for crimp of the carbon fibre weave. The mechanical properties in the glassy state were fitted to available experimentally determined data by using the properties of RTM6 in Table 1 and adjusting the crimp knock down factor, the in plane shear modulus and transverse poisson's ratio of the fibres in Table 3. To get an

10 estimate of the mechanical properties in the rubbery state the matrix properties in Table 2 and the fitted properties of the fibres in Table 3 were used.

Table 1. Properties of RTM6 in the glassy state. Matrix properties Source Symbol Value Unit

g Young’s modulus [17] Em 2.9 GPa υ g Poisson’s ratio Assumed m 0.35 - gg+υ g Shear modulus Emm2(1 ) Gm 1.1 GPa gg− υ g Bulk modulus Emm3(1 2 ) Km 3.2 GPa

α g -6 Coefficient of thermal expansion [23] m 69·10 1/°C Total volumetric chemical shrinkage [23] ∆VV 5.4 % ()∆ β g Coefficient of chemical shrinkage VV 3 m 1.8 %

Table 2. Properties of RTM6 in rubbery state. Matrix properties Source Symbol Value Unit

g r Shear modulus Gm 100 * Gm 0.011 GPa

α r -6 Coefficient of thermal expansion [23] m 136·10 1/°C ggαα r r Bulk modulus Kmm m[26] Km 1.6 GPa

r rr r+ r E Young's modulus 9(3)KGmm K m G m m 0.032 GPa

rr−+ rr υ r Poisson's ratio (32)(62)KGmm KG mm m 0.50 - β g β r Coefficient of chemical shrinkage m m 1.8 %

*: Assumed.

11 Table 3. Fibre properties. Fibre properties Source Symbol Value Unit Crimp knock down factor * η 0.89 -

f Longitudinal Young’s modulus [23] EL 238 GPa

f Transverse Young’s modulus [27] ET 20 GPa

f In plane shear modulus * GLT 20 GPa υ f Major Poisson’s ratio [27] LT 0.2 – υ f Transverse Poisson's ratio * TT 0.25 –

α f -7 Longitudinal coeff. of thermal expansion [23] L -1.00 10 1/°C

α f -5 Transverse coeff. of thermal expansion [23] T 1.00 10 1/°C

*: Adjusted to fit experimentally determined glassy properties of G986/RTM6 laminates.

Table 4. Mechanical laminate properties. Laminate properties* Glassy state Rubbery state Unit

Ex =Ey 44.1 38.9 GPa

Ez 8.4 3.0 GPa

νxy 0.32 0.33 -

νxz = νyz 0.28 0.55 -

Gxy 16.8 14.6 GPa

Gxz =Gyz 2.8 0.04 GPa -6 αx = αy 3.9 0.07 10 /°C -6 αz 65.3 191.8 10 /°C

-3 -5 βx = βy -9.4 10 -2.2 10 -

-2 -2 βz -1.5 10 -2.4 10 - *: E, ν,G,α and β are Young’s modulus, poisson ratio, shear modulus, coefficient of thermal expansion and coefficient of chemical shrinkage respectively. Process boundary conditions

The C-spar curing was modelled with a single cure step at 180°C until the maximum degree of cure was reached, which in the case of RTM6 epoxy is 94% according to [23]. The material model used to predict shape distortions [13] is

12 dependent on degree of cure and temperature and is independent of time. During the analysis it is assumed that the C-spar is cured under homogenous conditions and that the temperature and degree of cure are known. This means that the time scale used in the analysis is fictitious and that the cure schedule can be divided into a number of calculation steps as illustrated in Figure 6. The initial conditions were set to the conditions at the gel point where residual stresses and shape distortions start to form. This means an initial temperature of 180°Canda degree of cure of 60%. Then there is an isothermal cure until 94 % degree of cure is reached, followed by cooling of the component to room temperature. In

Table 5 the glass transition temperature, Tg has been calculated by Eqn. (8) using the degree of cure values presented in Figure 6. Four calculation steps are needed to capture the development of degree of cure, temperature and phase transitions during the cure process in a realistic way, see Table 5. The first calculation step covers the rubbery state. Then follows a small increment in glass transition temperature to simulate the rubber-glass transition (the vitrification [1]). Calculation step 3 describes the evolution of the degree of cure in the glassy state and finally, calculation step 4 represent the cooling from cure temperature to room temperature.

Table 5. Glass transition and cure temperatures at the end of each calculation step. Glass transition temp.*, Temperature,

Calculation step Tg (qC) T (qC) Material state

Initial 43.2 180 Rubbery, T < Tg 1 179.9 180 Rubbery, T < Tg 2 180.0 180 Glassy, T ≥ Tg 3 202.0 180 Glassy, T ≥ Tg 4 202.0 20 Glassy, T ≥ Tg *: Calculated from Figure 6 using Eqn (8).

13 1.00 200 0.95 180 160 (-) 0.90 (°C)

X 140 0.85 120 T 0.80 100 0.75 80 60 0.70

Degree of cure, 40 Temperature, 0.65 20 0.60 0 01234 Calcualtion step

Degree of cure Temperature Figure 6. Process description.

Mechanical boundary conditions

One realistic way of modelling the interaction between the component and the mould is by using contact conditions. However, in the present work it was decided to exclude the mould from the analysis and thereby the problems associated with contact. Instead two different boundary conditions will be evaluated to demonstrate the effect of the mould on shape distortion. In the first analysis the C-spar is considered free to move during all four calculation steps without influence from the mould. The only boundary condition imposed was to suppress rigid body motion. Secondly, an analysis was performed where all nodes were constrained during the four calculation steps, simulating the interaction between the mould and the C-spar. To simulate demoulding a fifth calculation step was added where the boundary conditions were changed to suppress rigid body motion only. Figure 7 shows the boundary conditions used to suppress rigid body motion. All three nodes in Figure 7 were constrained in the local 3-direction, node 2 and 3 were also constrained in the local 2-direction and finally node 3 was in addition constrained in the local 1-direction.

14 Global

Local Node 3. Node 1. Node 2.

Figure 7. Boundary conditions to suppress rigid body motions, the black dots represent constrained nodes.

Results

In Figure 8 a plot of the predicted overall deformation behaviour of the C-spar is shown. The following deformation behaviour could be identified:

• The cleats and the closed end restrain deformation.

• A local spring-in occurs between the cleats.

• A “global” spring-in occurs i.e. the spar is bending inwards.

• An “out-of-plane” displacement can be seen i.e. a bending upwards in the open u-direction.

• A twist about the axis in the length direction of the spar.

• An overall shrinkage in the length direction can be seen.

15 Manufacturing induced deformations (Displacement magnification factor = 10)

Undeformed (tool)

Figure 8. Global deformations of the C-spar.

C D

B E θ θ Inner angle, i Outer angle, o

A Undeformed F Deformed ∆θ ∆θ i o

Inner spring-in Outer spring-in Figure 9 Cross-section with schematic illustration of spring-in.

To be able to make a comparison between the experimentally observed and predicted shape distortions, the predicted spring-in was calculated at cross- sections 1, 3, 4, 5 and 7 defined in Figure 4. In Figure 9 a cross-section of the C- spar is shown. For each cross-section the undeformed and deformed coordinates of six nodes, represented by the marks (A-F), was used to calculate the spring- ∆θ ∆θ in. Outer, o and inner spring-in, i were calculated using the definition of the scalar product,

16 uv⋅= u ⋅ v ⋅cosθ (9) For inner spring-in the vector u is a vector between point B and point A and vector v is a vector between point C and point D. The difference between the θ inner C-spar angle, i , using undeformed and deformed coordinates is the inner ∆θ spring-in angle, i , for the particular cross-section. The outer spring-in was calculated in a similar way, where u was a vector between point E and point F and v was a vector between point D and point C. Predicted and experimental inner and outer spring-in, at the different cross- sections defined in Figure 4, are presented in Figure 10 and Figure 11. The predictions using the simulation tool are based on the two different boundary conditions described in the previous section. As a reference, Figure 10 and Figure 11 contain predictions of spring-in where only the thermal effect is accounted for, i.e. the composite is assumed stress free at the cure temperature and the associated shape distortions are formed during the cooling without influence of the mould. The reference analysis was performed as a step change in temperature from 180 to 20°C with a boundary condition that only suppressed rigid body motion, see Figure 7. Finally, the experimental results are based on measured shape distortions on three manufactured C-spar’s.

1.60

1.40

1.20 (°)

i 1.00 'T 0.80

0.60 Spring-in, 0.40

0.20

0.00 01234567 Cross section

Predicted, Free Predicted, Constrained Ref. Pred., Thermal, Free Experiments Figure 10. Inner spring-in, 95% confidence limits on the experimental results is based on pooled standard deviation. Experimental results used with permission from BAESYSTEMS on behalf of the PRECIMOULD consortium.

17 1.60

1.40

1.20 (°) 1.00 'TR 0.80

0.60 Spring-in, 0.40

0.20

0.00 01234567 Cross section

Predicted, Free Predicted, Constrained Ref. Pred., Thermal, Free Experiments Figure 11. Outer spring-in, 95% confidence limits on the experimental results is based on pooled standard deviation. Experimental results used with permission from BAESYSTEMS on behalf of the PRECIMOULD consortium.

Discussion

By comparing the predicted and experimental results in Figure 10 and Figure 11 it is obvious that using the simulation tool with the constrained boundary condition shows best agreement both for the inner and outer spring-in. Using the simulation tool with free boundary condition overestimate the shape distortions and if only the thermal effect is considered, as in the reference analysis, the distortions are underestimated. During cure in a stiff mould residual stresses are developed and shape distortions are formed at demoulding by release of macro scale residual stresses. If the effect of the mould is neglected in the analysis shape distortions are instead formed during the entire cure schedule without development of macro scale residual stresses. As shown in Figure 10 and Figure 11 a specimen free to move during cure is predicted to exhibit a larger shape distortion than a constrained specimen. The reason is that, according to the material model used, Eqn. (4), deformations are frozen into the material at the rubber–glass transition during constrained cure [15]. The frozen deformations are released if the material is reheated freestanding above the glass transition temperature. The reason for the good agreement between the predicted results using the constrained boundary condition and the experimental results suggests that the C-

18 spar is well bonded to at least one of the tool surfaces until after vitrification. In a previous work [15] (using different materials and mould) negligible adhesion between the mould and composite was obtained by placing mylar films at the interfaces. For that case constraining all nodes, as was done in this work, resulted in a too high degree of constraint compared to reality. The experimental results were in between predictions based on free and constrained boundary conditions and matched predictions based on contact boundary conditions. This is in accordance with results by Fernlund et al. [9] who investigated spring-in of C-channels manufactured by carbon/epoxy prepregs on an aluminium tool. They found that the spring-in was 20% less for a tool with no release agent as compared to a tool with release agent. The details of the interaction between the mould and component are an area for further investigation and beyond the scope of the present paper. Another observation is that the experimental results show a significant difference between the average inner and outer spring-in. The difference is most pronounced at cross-section 1 where the average experimental outer spring-in was 0.97° as compared to the inner spring-in of 0.65°, a difference of 0.32°. Also the predictions give a significant difference between outer and inner spring-in for cross-section 1, 0.24° when the constrained boundary conditions was used. The experiments results also indicate a small difference for the other cross-sections, which the predictions do not. However, comparisons between predicted and experimental results show that the simplified constitutive model, the estimated mechanical properties, the simulation approach and the constrained boundary conditions used here give a good estimate of the shape distortions of a structural component manufactured by a typical RTM process cycle. An advantage, at least for industrial use, is that the simulation tool and approach presented here is not computer intensive and require a minimum of material characterisation as compared to fully viscoelastic process models. The problems presented here took about one hour to solve using a computer with a 350 MHz PENTIUM II processor equipped with 256 MB of RAM.

Conclusions

In the present paper a simplified constitutive model and simulation approach, for prediction of shape distortions, previously presented in [13, 15] have been further validated using a different material and geometry. The feasibility to simulate large parts of complex shape has also been verified. The results show, for the material and manufacturing process considered here, that the simulations give a good estimate of cure induced shape distortions, even if a number of assumptions and simplifications have been used, such as

19 • a simplified constitutive model, • assumed isothermal homogenous curing conditions, • idealised cure schedule, • laminate material properties estimated from the constituent materials, • a simplified interaction between the mould and the component

Simulations based on different mechanical boundary conditions give significantly different results and in this case the best results were obtained by considering the part as bonded to the mould.

Acknowledgements

The authors gratefully acknowledge BAESYSTEMS on behalf of the PRECIMOULD consortium for making the C-spar geometry and the experimental results available. This work is a part of a national aeronautic research program (NFFP) funded by the Defence Materials Administration in Sweden (FMV). Additional financial support has been achieved from The Swedish Foundation for Strategic Research through the Integral Vehicle Structure research school (IVS), and IRECO.

References

1. Aronhime MT, Gillham JK. Time-Temperature-Transformation (TTT) Cure Diagram of Thermosetting Polymeric Systems. In: Dusek K., editor. Epoxy Resins and Composites III, Berlin: Springer-Verlag, 1986:83-113. 2. Radford DW, Rennick TS. Separating Source of Manufacturing Distortion in Laminated Composites. Journal of Reinforced Plastics and Composites 2000;19(8):621-641. 3. Bogetti TA, Gillespie JH. Process-Induced Stress and Deformation in Thick- Section Thermoset Composite Laminate. Journal of Composite Materials 1992;26(5):626-660. 4. White SR, Hahn HT. Process Modeling of Composite Materials: Residual Stress Development during Cure, Part I Model Formulation. Journal of Composite Materials 1992;26(16):2402-2422. 5. White SR, Hahn HT. Process Modeling of Composite Materials: Residual Stress Development during Cure, Part II Experimental Validation. Journal of Composite Materials 1992;26(16):2423-2453. 6. White SR, Kim KK. Process-Induced Residual Stress Analysis of AS4/3501-6 Composite Material. Mechanics of Composite Material and Structures 1998;5:153-186.

20 7. Prasatya P, McKenna GB, Simon SL. A Viscoelastic Model for Predicting Isotropic Residual Stresses in Thermosetting Materials: Effects of Processing Parameters. Journal of Composite Materials 2001;35:826-849. 8. Johnston A, Vaziri R, Poursartip A. Journal of Composite Materials 2001; 35: 1435-1469. 9. Fernlund G, Rahman N, Courdji R, Bresslauer M, Poursartip A, Willden K, Nelson K. Experimental and numerical study of the effect of cure cycle, tool surface, geometry, and lay-up on the dimensional fidelity of autoclave- processed composite parts. Composites Part A: Applied Science and Manufacturing 2002;33:341-351. 10. Hibbit, Karlsson & Sorensen Inc. ABAQUS/Standard User’s manual, Version 5.8, 1998. 11. Poon H, Koric S, Ahmad MF. Towards a Complete Three Dimensional Cure Simulation of Thermosetting Composites. Proc. Int. Conf. on ‘Advanced Composites’, Hurghada, Egypt, December 1998, 165-180, Editors: Y. Gowayed, Department of Textile Engineering, Auburn University, Alabama, USA and F. Adb El Hady, Department of Automotive Engineering, Ain Shams University, Cairo, Egypt 12. Huang X, Gillespie JW, Bogetti T. Process induced stress for woven fabric thick section composite structures. Composite Structures 2000;49:303-312. 13. Svanberg JM, Holmberg JA. Predictions of shape distortion, Part I FE- implementation of a path dependent constitutive model. Submitted to Composites Part A: Applied Science and Manufacturing, Paper B in this thesis. 14. Svanberg JM, Holmberg JA. An experimental investigation on mechanisms for manufacturing induced shape distortion in homogenous balanced laminates. Composites Part A: Applied Science and Manufacturing 2001;32(6):827-838, Paper A in this thesis. 15. Svanberg JM, Holmberg JA. Predictions of shape distortion, Part II Experimental validation and analysis of boundary conditions. Submitted to Composites Part A: Applied Science and Manufacturing, Paper C in this thesis 16 Svanberg JM. Shape distortion of a non-isothermally cured composite angle bracket. Accepted for publication in Plastics Rubber and Composites, Paper D in this thesis. 17. Product Data Sheet: RTM 6. HEXCEL Composites, 1998. 18. Holmberg JA, Influence of Chemical Shrinkage on Shape Distortion of RTM Composites, Proc 19th International SAMPE European Conference of

21 the Society for the Advancement of Material and Process Engineering, Paris, France, 22-24 April, 1998:621-632. 19. Christensen RM. Theory of viscoelasticity, 2nd edn, 1982, New York, Academic Press. 20 Nielsen LE. Cross-Linking-Effects on Physical Properties of Polymers. J. Macromol. Sci. –Revs. Macromol. Chem. 1969;C3(1):69-103. 21 Simon SL, McKenna GB, Sindt O. Modeling the Evolution of the Dynamic Mechanical Properties of a Commercial Epoxy During Cure after Gelation. Journal of Applied Polymer Science 2000;76(4):495-508. 22 Zocher MA, Grooves SE and Allen DH. A Three-Dimensional Finite Element Formulation for Thermoviscoelastic Media. International Journal for Numerical Methods in Engineering 1997;40:2267-2288. 23. Holmberg JA. Resin Transfer Moulded Composite Materials. Doctoral Thesis, Luleå University of Technology, Department of Materials and Manufacturing Engineering, Division of Polymer Engineering, Report 1997:10, 1997. 24. Whitney JM, McCullough RL. Micromechanical Material Modelling, Delaware Composites Design Encyclopedia-Volume 2. 1990, 65-72, Technomic Pub. Co. Inc., Lancaster, Pennsylvania, USA 25. Gudmundsson P, Zang W. An Analytic Model for Thermoelastic Properties of Composite Lamiantes Containing Transverse Matrix Cracks. International Journal of Solids and Structures 1993;30:3211-3231. 26. Plepys AR, Farris RJ. Evolution of residual stresses in three-dimensionally constrained epoxy resin. Polymer 1990;31:1932-1936. 27. Hull D. An introduction to composite materials, Cambridge: University Press, 1988.

22 2002:40 1402-1544 LTU-DT--02/40--SE

Utbildning Doctoral thesis Institution Upplaga Tillämpad fysik, maskin- och materialteknik 250 Avdelning Datum Polymerteknik 2002-10-22 Titel Predictions of manufacturing induced shape distortions -high performance thermoset composites Författare Språk J. Magnus Svanberg Engelska Sammanfattning High performance composites usually consist of continuous fibres and a thermoset matrix. A well-known example is carbon fibre epoxy composites. When this kind of material is cured residual stresses and/or shape distortions are produced owing to thermally and chemically induced volumetric strains. The cure means the manufacturing step where the thermoset matrix is transformed from a liquid to a solid material. It is a quite complex thermal- chemical- mechanical process that in addition to volumetric strains, involves heat generation and dramatic changes in mechanical properties. For manufacturing of parts with high shape tolerances, such as aircraft components, the geometry of the mould is compensated to accommodate for shape distortions. Today this is made based on thumb rules and experience followed by trials. This is time consuming and expensive. Development of a tool for prediction of shape distortions and residual stresses is therefore an important step towards more optimised manufacturing of composites. The present thesis, consisting of five papers, describes the development and validation of a simulation tool for prediction of shape distortion and residual stresses. In the first paper a typical material and manufacturing process for high performance composites was used to experimentally investigate the effects from the cure temperature on spring-in of angle sections. The experimental results were interpreted in terms of mechanisms .....(cont.)

Granskare/Handledare Janis Varna

URL: http://epubl.luth.se/1402-1544/2002/40