Predicting the process-dependent material properties to evaluate the warpage of a co-cured epoxy-based composite on metal structures

Composites Part A 133 (2020) 105857

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Predicting the process-dependent material properties to evaluate the warpage of a co-cured epoxy-based composite on metal structures

T

⁎

Robert Thomas , Simon Wehler, Fabian Fischer Volkswagen AG – Group Innovation, Postbox 011/1777, 38440 Wolfsburg, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: A. Thermosetting B. Cure behaviour C. Thermomechanical E. Process Simulation

By directly curing continuous fiber reinforced thermosets locally onto metal parts, the mechanical performance of these components can be increased significantly by just adding a minimum of weight. Due to the thermal and chemical behavior of the composite material, residual stresses occur and lead to warpage of the structure or failure of the part. This paper deals with a hot-melt, snap-cure matrix material which is characterized as a function of temperature and degree of cure, also the cure kinetics are described based on an Arrhenius-equation. This information is used to simulate the curing of a composite for a press-molding manufacturing process in order to calculate the temperature profile of the material, taking the exothermal reaction of the epoxy into account. Also a metal-composite-hybrid specimen is simulated to evaluate the influence of the manufacturing process on the warpage of the component. Manufacturing trials evaluate the simulation and allow an estimation regarding the forecast quality. Finally, a parametric study illustrated the influence of selected temperature- and degree of cure dependent characteristic values on the prediction accuracy of the simulation to outline the significant influencing factors.

1. Introduction Present body-in-white (BiW) structures of high-volume car models are mainly made out of metallic materials. For crash-loaded components (i.e. sill section, bumper cross beam), high-strength or hot formed steel types are used to absorb the resulting high energy. Because of the extensive integration of batteries into the floor structure of batteryelectric vehicles, in order to increase the range, the crash weight and consequently also the crash load of these cars is increasing [1–3]. Examples include side pole impact or full width frontal impact [4,5]. In addition to the acceleration values of the occupant, the intrusion of the pile and adjoining components into the vehicle structure is evaluated. The acceleration is predominantly reduced by passive safety systems like seat belt tensioner or various airbags, but the intrusion length must be reduced by the vehicle structure itself. Here, conventional metallic solutions reach their limits, but by combining metal structures locally with continuously fiber reinforced plastics (frp), innovative and efficient solutions can be established [6,7]. The local application of the composite in highly loaded car sections ensure a minimum of weight increase in regard to metallic solutions. Also conventional BiW-joining technologies, like welding or riveting can be maintained, if the joining area is kept free of the plastic reinforcement. Previous studies have shown, that for these thermoset materials curing times of under 60 s can ⁎

be utilized, which enables an application in a high-volume car production, but increased process temperatures of up to 180 °C have to be applied [8,9]. The origin of residual stresses, which can cause a deformation of composite parts can be categorized into intrinsic and extrinsic sources. Intrinsic sources are defined by different thermal expansions of the base materials or chemical shrinkage of the matrix material. Extrinsic sources are defined by the layup of the composite or part- and tooling geometry.[10,11] In this work, a thermoset prepreg material is cured onto a metallic joining partner at increased temperatures and subsequently cooled down to room temperature. Due to the different thermal expansion coefficients of the joining partner and the chemical shrinkage of the matrix material, residual stresses occur during the manufacturing process [10]. Moreover, these stresses can be divided into micro-mechanical and macro-mechanical. Micro-mechanical residual stresses occur because of the interaction between fiber and matrix, i.e. in one layer of composite and subsequently lead to longitudinal and transversal intralaminar tension. This micro-mechanical stress has an influence on the strength of the composite (because of this initial load) but not on the deformation of the part.[10] Macro-mechanical residual stresses occur between different layers of composite or different materials due to their varying elongation (resulting from different thermal expansion or chemical shrinkage). This results into interlaminar stresses

Corresponding author. E-mail address: [email protected] (R. Thomas).

https://doi.org/10.1016/j.compositesa.2020.105857 Received 21 June 2019; Received in revised form 12 February 2020; Accepted 25 February 2020 Available online 27 February 2020 1359-835X/ © 2020 Elsevier Ltd. All rights reserved.

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in longitudinal and transversal direction which vary along the thickness of the part due to different fiber orientations or part geometries. In consequence these residual stresses are relaxing by deformation of the manufactured part.[10] The aim of this paper is to determine this deformation along the manufacturing process. Early researches deal with the determination of the warpage during the cooling process by using incremental linear elasticity [12]. Later on, also the chemical shrinkage was considered for the calculation of the residual stress and the resulting part deformation in composite parts [13]. By calculating the visco-elastic effects of the material, which describe the stress relaxation in the loaded part, an increase of prediction accuracy can be achieved, but uses exceeded computational resources [14,15]. Also several current researches evaluated the residual deformation of composites by using different approaches and showed a promising correlation between numerical simulation and the experiment for different manufacturing technologies [16–19]. But these examinations were made for pure thermoset composites with slow-curing matrix materials with cycle times above 1 h. A major influence on the occuring deformation in the manufacturing process was found in the curing temperature of the composite. A resin transfer moulded (RTM) part with a curing time of 7 min was cured at different temperatures. Higher curing temperatures lead to an increased warpage of the part because of the increased thermal elongation when cooling down to room temperature. An analytical approach based on the chemical shrinkage and the different thermal expansions was introduced, whereas constant parameters were used. The comparison of experiment and calculation showed an insufficient matching.[20] Therefore, this paper is built on these preliminary researches and comprehensively considers the curing- and cooling process of a metal-composite hybrid part for a fast-curing epoxy matrix material with cycle times below 1 min to evaluate the cure speed, temperature development in the composite and emerging warpage of the structure. To predict the behavior of the metal-composite hybrid specimen through the manufacturing process, the material properties and their change while curing have to be described as a function of degree of cure and temperature. In this paper, firstly the reaction kinetics was determined, in order to calculate the reaction speed and heat of reaction for various temperature profiles. By coupling the thermal behavior of the material with the chemical alternation of the characteristic values, also an exothermal self-heating on part level could be considered. These results were compared with molding trials of composite sheets. By additionally coupling the mechanical with the thermal-chemical properties, structural effects on the substructure were determined and confirmed on a metal-composite-hybrid specimen. The determination of these necessary material properties are timeand cost-intensive and should be kept to a minimum for upcoming development projects. Therefore the relevant characteristic values influencing the accuracy of the simulation results were identified by means of a sensitivity analysis. These results along with the consideration of a fast-curing matrix material and the mathematical description of its material properties as a function of degree of cure and temperature represent the novelty of this work.

Table 1 Extract from the mechanical and thermal characteristics of glass fiber and metal [21–23]. description

symbol

characteristic value

unit

density of glass-fiber density of metal specific heat capacity of glass fiber

ρf ρmetal cp,f cp,metal

2, 54 7, 85 0, 96

g/cm3 g/cm3 kJ/kg K

0, 49 1, 22

kJ/kg K

λ f‖ λ f⊥

1, 22

W/mK

λmetal Ef Emetal νf νmetal

44, 5 73 200 0, 22 0, 3

α fth

5, 1·10−6

W/mK GPa GPa – – 1/K

th α metal

12, 3·10−6

1/K

specific heat capacity of metal thermal conductivity of glass fiber parallel thermal conductivity of glass fiber transversal thermal conductivity of metal Young’s modulus of glass fiber Young’s modulus of metal Poissons-ratio of glass fiber Poissons-ratio of metal coefficient of thermal expansion of glass fiber coefficient of thermal expansion of metal

W/mK

3. Description of material properties Following, the approach to describe the effects during manufacturing and the necessary material characterization is outlined. Since the characterized data will be used for a downstream process simulation, also a mathematical fit to continuously describe the course of the characteristic value is presented. 3.1. Reaction kinetics Modelling the curing process of fiber reinforced materials has been the topic of many preceded scientific papers. There are two different approaches to describe the reaction kinetics – the model-fitting and the model-free or iso-conversional approach, where both can be described by an Arrhenius equation.

dα E = k (T ) f (α ) = Aexp ⎛− a ⎞ f (α ) dt ⎝ RT ⎠ dα dt

(1)

So, the reaction rate is defined by the product of the temperaturedependent rate constant k (T ) and the reaction model f (α ) . In turn the rate constant is defined by the frequency factor (also called pre-exponential factor) A, the activation energy Ea , the universal gas constant R and the temperature T. Ea, A and f (α ) are occasionally called the kinetic triplet [24]. Model-fit-methods use existing approaches to imitate the curing behavior, the activation energy and frequency factor are constant [25]. For iso-conversional models, the activation energy is defined as a function of degree of cure and it is assumed that the reaction model is independent of temperature and heating rate [26,27]. The model-fitting-method in combination with isothermal data has shown a good relation between experiment and simulation [28], but is not applicable for non-isothermal processes [29–32]. The considered matrix material of this work, manufactured in a press-molding process, cures within seconds, so an isothermal characterization by using differential scanning calorimetry (DSC) could not be realized. In consequence, nonisothermal DSC-tests had to be performed and consequently an isoconversional approach was used to calculate the cure kinetics. It is known, that this method is appropriate to describe the curing, but the transferability regarding to fast curing matrix materials must also be verified. Non-isothermal DSC-trials on a Netzsch 204 F1 Phoenix with constant heating rates of β = (5, 10, 15, 20 and 30) K/min were performed (see Fig. 1). As expected, the DSC-signal and so the exothermal reaction heat increases with higher heating rates, same applies to the starting

2. Materials A pre-impregnated fast-curing unidirectional continuous glass fiber reinforced epoxy material (prepreg) with a fiber volume content of 45,5 % and a layer thickness of 0,65 mm was used. The metallic joining partner for the metal-frp-hybrid specimen was defined as a galvanized and oiled dual phase steel with a thickness of 1,6 mm. The determination of matrix-relevant characteristic values was performed with the matrix material itself. The mechanical and thermal characteristics of the fiber and metal were specified by use of parameters gained through literature research, and can be found in Table 1.

2

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Fig. 1. DSC trials for different heating rates.

Fig. 2. Activation energy Ea as a function of degree of cure α .

temperature of the reaction. To calculate the reaction rate the following expression can be used [33].

βi (i = 1…n ) [39–41], the determined sets of non-isothermal DSC-runs can be expressed with Eq. (10)

dα H (t ) = dt ΔHtotal

g (α ) =

(2)

n

n

∑∑ i=1

(3)

− as tangent in the beginning of S(t ), − as tangent in the end of S(t ).

t

S (t ) − B (t ) dt

j≠i

I(Ea (α ), Tα, j ) βi

Φ(Ea (α )) =

dT dt

(6)

ln(A) = aEa + b

α

1 A dα = f (α ) β

g (α ) =

∫0

α

∫0

T

(7)

Ea ⎞ exp ⎛− dT , ⎝ RT (t ) ⎠ ⎜

= n (n − 1). (11)

j≠i

I(Ea (α ), Tα, i ) βj I(Ea (α ), Tα, j ) βi

(12)

(13)

The intersect of the resulting straight lines define ln(A0 ) and Ea,0 , compare Fig. 3. ln(A0 ) and Ea,0 are independent of the heating rate and can thus be used to calculate a and b and consequently to determine the frequency factor A (see Fig. 4). With the known Ea (α ) and A (α ) the reaction model f (α ) can be calculated by transposing Eq. (7).

⎟

An expedient approach for describing iso-conversional kinetics can be presented with a nonlinear temperature integral [31,36–38]. So Eq. (7) results in

∫0

(10)

The result is shown in Fig. 2. There are several approaches for the evaluation of the frequency factor A. This work uses the compensation parameter method. Preceded research papers have shown that the parameters of the model-fit-approach are dependent of the heating rate, but are linked by the compensative effect, which can be expressed with Eq. (13), where a and b are the parameters of the linear regression [43,44].

Combining Eq. (6) with Eq. (1) the following expression results.

k (T ) f (α ) dα A −Ea ⎞ f (α ) = = exp ⎛ dT β β ⎝ RT (t ) ⎠

n

∑∑ i=1

(5)

0

⎜

Tα, n ).

(4)

The heating rate β is the defined by the change of temperature depending on the time.

β=

=

I(Ea (α ), Tα, i ) βj

n

∫t

Tα,2)

A (α ) I(Ea (α ), βn

The indices stand for the different heating rates and their related temperature profile. The activation energy Ea as a function of degree of cure α is determined by fitting the value of Ea which minimizes the following function (see Eq. (12)) [29,31,42].

This iteration was terminated when the alternation between two iteration steps fell below 10−6 . With the following equation the overall enthalpy for the selected heating rates was calculated (see Table 2).

ΔHtotal =

A (α ) I(Ea (α ), β2

therefore follows

with

a1 + b1t a2 + b2 t

Tα,1) =

=…

H (t ) is the released enthalpy at time t and ΔHtotal the overall enthalpy of the non-cured matrix material. During the non-isothermal DSC-measurement effects like variation of specific thermal capacity and their dependency to the temperature have an influence on the DSC-signal S (t ) . To compensate these for calculating the enthalpy of the material, a modified baseline B (t ) was iterated [34,35]. B(t ) = (1 − α (t ))(a1 + b1t ) + α (t )(a2 + b2 t )

A (α ) I(Ea (α ), β1

⎟

(8)

1 A dα = I(Ea (α ), Tα ). f (α ) β

(9)

By assuming, that the reaction model is independent of the heating rate Table 2 Measured enthalpy of reaction for different heating rates. heating rate β in K/min 5 10 15 20 30

enthalpy of reaction ΔHtotal in J/g

342, 341, 338, 340, 340,

30 10 44 76 74

± ± ± ± ±

2, 1, 5, 1, 0,

34 13 45 42 59

Fig. 3. Compensative effect for the different heating rates and evaluation of ln(A0 ) and Ea,0 . 3

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hybrid parts, this exothermal effect has to be considered. Generally the heat flow can be described by the energy intake and outlet and the generated energy (e.g. by exothermal reaction).

̇ + Qgen ̇ = Q̇in − Qout

ΔQ Δt

(16)

By using the Fourier-approach, the equation can be expressed as [45]

∂T ⎞ ̇ = ρcp ⎛→ ▿ ·(λ ▿T ) + Qgen V · ▿T + . ∂t ⎠ ⎝

(17)

The evaluated composite consists of the unidirectional glass fibers enclosed with the reactive epoxy material. To calculate the thermal energy, several assumptions were introduced. A pre-impregnated fabric is used, so a significant flow of material during the press process can be → excluded and the velocity tensor results to V = 0 . The heat transfer between fiber and matrix won’t be considered, so an instant equilibrium is assumed. Consequently the material parameters of the composite can be calculated by use of the rule of mixture. Also a transversal-orthotropic laminate is assumed – here is no linking between normal strain and shear distortion [22]. Effects which may be caused by voids or humidity are excluded. Therefore Eq. (17) can be transposed to Eq. (18).

Fig. 4. Frequency factor A as a function of degree of cure α .

ρtotal cp,total

Fig. 5. Reaction model f (α ) as a function of degree of cure α .

∂T = ∂t

λ x ,total

( ) ( ) ( ) + Q̇ ∂2T ∂x 2

+ λ y,total

∂2T ∂y 2

+ λ z,total

∂2T ∂z 2

gen

(18)

−1

dα E (α ) ⎞ ⎤ f (α ) = β ⎛ ⎞ ·⎡A (α )exp ⎛− a ⎝ dT ⎠α ⎢ ⎝ RT (α ) ⎠ ⎥ ⎣ ⎦ ⎜

ρtotal describes the resulting density, cp,total the resulting heat capacity and λ x ,total , λ y,total , λ z,total the thermal conductivity of the composite. The generated heat flow based on the exothermal reaction of the matrix material can be implemented by the following equation [13]

⎟

(14)

The course of the reaction model depending on the degree of cure is shown in Fig. 5. Finally the kinetic triplet is completely described and the degree of cure as a function of time can be calculated with

α=

∫t

tEnd

0

E (α ) ⎞ A (α )exp ⎛− a ·f (α ) dt . RT (t ) ⎠ ⎝ ⎜

̇ = (1 − φ) ρm H · Qgen

dα , dt

(19)

with the fiber-volume content φ , the density of the epoxy ρm , the enthalpy of reaction of the matrix material H and the reaction rate of the curing dα . The curing process affects the structure of the matrix matedt rial, monomers cross-link to form complex polymer chains and the material changes from liquid to solid. Furthermore the properties of the epoxy vary with the temperature. So, if possible, these characteristic values were determined as a function of degree of cure and temperature. The comparative procedure was applied to measure the density of the matrix material using a precisions scale and ethanol as calibration liquid. Studies exist, where the density was characterized as a function of degree of cure [46–48]. This was not feasible for the considered epoxy material due to the increased curing temperature and curing speed. In consequence, the matrix density was determined at α = 0 and α=1 ρm (α = 0) = (1, 143 ± 0, 003) g/cm3 with and ρm (α = 1) = (1, 211 ± 0, 003) g/cm3 . The change of density as a function of temperature due to the thermal expansion was considered by the thermal expansion coefficient, which is explained later. A linear correlation between ρm and α was presumed, which should be an adequate adaption to the actual dependency [49].

⎟

(15)

To validate this mathematical approach, the results of the DSC-trials were predicted (see Fig. 6a) and a reasonable consistency was shown. Furthermore the evolution of the degree of cure for a defined temperature profile is demonstrated in Fig. 6b. 3.2. Thermo-chemical coupling Due to the low mass of the DSC-specimen, it is assumed that a minimal exothermal self-heating occurs during the measurement. This assumption changes when manufacturing composite parts with a certain thickness. Exothermal reaction leads to an increase of the temperature and thus to an accelerated curing and vice versa. Furthermore a temperature gradient for thicker structures can result. For evaluating the curing process and resulting residual stresses of metal-composite-

ρm (α ) = 0, 065 g/cm3·α + 1, 145 g/cm3

(20)

To homogenize this matrix density to the composite density, the following relation was used.

ρtotal (α ) = ρf φ + (1 − φ) ρm (α )

(21)

The glass transition temperature is described as a temperature, where the matrix material reversibly changes its condition (e.g. from hard to rubbery when increasing temperature at α = 1). This glass transition temperature is dependent of the degree of cure and can be calculated by

Fig. 6. Evaluation of reaction kinetics approach. 4

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Table 3 Glass transition temperature TG measured for several degrees of cure α . degree of cure α 0 0, 16 ± 0, 34 ± 0, 61 ± 0, 82 ± 1

Table 4 Specific heat capacity cp,m determined for different degrees of cure α .

glass transition temperature TG in °C

degree of cure α

specific heat capacity cp,m (α ) in kJ/kg K

6, 53 ± 0, 18 21, 20 ± 0, 57 32, 15 ± 0, 10 47, 75 ± 1, 10 61, 45 ± 0, 35 125, 10 ± 1, 17

0 0, 31 ± 0, 028 0, 44 ± 0, 035 0, 61 ± 0, 014 1

4, 34 ± 0, 35 3, 92 ± 0, 025 3, 52 ± 0, 012 3, 26 ± 0, 082 2, 22 ± 0, 091

0, 04 0, 05 0, 02 0, 01

the DiBenedetto-equation [50].

TG (α ) − TG0 λTG ·α = TG1 − TG0 1 − (1 − λTG)·α

(22)

TG0 and TG1 describe the glass transition temperature for an unreacted and a fully cured matrix material, λTG is defined as a fitting parameter and α characterizes the degree of cure. Specimen with different degrees of conversion were manufactured and the glass transition temperature was determined by using a DSC, see Table 3. The fitting parameter λTG was calculated with λTG = 0, 32 , the comparison between the measured glass transition temperatures and the DiBenedetto-approach is shown in Fig. 7. The specific heat capacity is dependent on the degree of cure and the temperature. To estimate the impact of the degree of cure on the specific heat capacity, specimen with different degrees of conversion were manufactured and the specific heat capacity was measured. The measuring range was defined with respect to the glass transition temperature, which can be calculated with Eq. (22). The maximum temperature was set below this value for the specific specimen to prevent additional curing effects. The results of these experiments are listed in Table 4. The results show a linear relation (see Fig. 8) which can be calculated by the following equation.

cp,m (α ) = −2, 144 kJ/kg K·α + 4, 465 kJ/kg K

Fig. 8. Comparison of measured and calculated specific heat capacity as a function of degree of cure.

(23)

The dependency of the specific heat capacity as a function of temperature was measured on fully cured specimen, the results are shown in Fig. 9. Three significant sections are apparent. In the beginning, a linear increase of the specific heat capacity was identified which can be described with the following expression.

cp1,m (T ) = 0, 0057 kJ/kg

K2 · T

+ 1, 1 kJ/kg K

Fig. 9. Comparison of measured and calculated specific heat capacity as a function of temperature for a fully cured specimen.

cp3,m (T ) =

1 + exp

(24)

Within the area of glass transition, the specific heat capacity increases significantly and then runs into a plateau.

cp2,m (T ) = 2, 2 kJ/kg K

cp2,m (T ) − cp1,m (T )

(

TG1 − T cc

)

+ cp1,m (T ) (26)

The constant cc = 5, 96 was calculated by using a least squares fit method. For T ≪ TG1, the denominator leads towards infinity, so cp3,m (T ≪ TG1) = cp1,m (T ) . For T ≫ TG1, the denominator leads towards 1 and the Eq. (26) results in cp3,m (T ≫ TG1) = cp2,m (T ) . For temperatures close to TG1, a transition between the two characteristic courses is established. The specific heat capacity is now defined separately as a function of degree of cure and temperature, the link between both equations is managed by a Heaviside step function U(x ) ,

(25)

The transition area between the linear course and the plateau is dependent of the glass transition temperature and can be characterized with Eq. (26) [19].

cp,m, t (α, T )= [cp,m (α )·(1 − U(α − 1)] +

[cp3,m (T )·U(α − 1)],

(27)

with

0, for α < 1, U(α − 1) = ⎧ ⎨ ⎩1, for α ⩾ 1.

(28)

The Eq. (28) describes the specific heat capacity of the matrix material. The evolution of the specific heat capacity for a defined temperature profile (see Fig. 6b) is demonstrated in Fig. 10. The specific heat capacity of a unidirectional composite cp,UD (α, T ) can be described with following approach [22].

Fig. 7. Comparison of the measured glass transition temperature with the DiBenedetto-approach. 5

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⎡ cos2 (θ) sin2 (θ) − 2 sin(2θ) ⎤ ⎢ ⎥ 1 T = ⎢ sin2 (θ) cos2 (θ) sin(2θ) ⎥ 2 ⎢ ⎥ ⎣ sin(2θ) − sin(2θ) cos(2θ) ⎦

(35)

For a laminate with various layers, the total thermal conductivity can be calculated with n

λ x total =

∑

λ xk · h

hk

λ yk · h

hk

total

k=1 n

λ ytotal =

λ z total =

(29)

The specific heat capacity of the fiber cp,f is assumed as constant. For laminates with different fiber orientations, the Eq. (30) can be used, where h defines the thickness of the layer and the index k the properties of the layer k for n layers of the laminate. n

(30)

As a reinforcement, a continuous glass fiber was used for the trials of this paper, which has an isotropic structure, consequently λ f‖ = λ f ⊥. For a unidirectional layer of a composite, the specific heat capacity in the direction of fiber λ‖ can be calculated with Eq. (32) [22]. (32)

Transversal to the direction of fiber, a physically based approach according to Rayleigh was used, which has shown good correlation with results of experiments [52–54].

2φ λ⊥ =1− C λm ν′ + φ − ν 1′ φ4 −

n

∑i = 1 Em (α ) =

C2 8 φ ν′

E (i) (i)

(38)

− Em1

a −α⎞ 1 + exp ⎜⎛ ⎟ (i) ⎝ c ⎠ αGel − α

(33)

(

c (i + 1)

)

+ Em1 (39)

The index i defines the count of iterations until obtaining a sufficient accuracy. For that, a Spearman’s rank correlation coefficient of R2 ⩾ 0, 95 was used [55]. The missing coefficients of Eq. (38) are calculated by a non-linear method of the least-squares fit method, see Table 6. A comparison between the measured and calculated modulus of the matrix material depending on the degree of cure is shown in Fig. 11.

C1 = 0, 3058 C2 = 0, 0134 λm +1 λ⊥ f λm −1 λ⊥ f

To consider the orientation of the layer, a transformation from local to global coordinate system has to be performed by use of Eqs. (34) and (35) [22].

⎡ λx ⎤ ⎡ λ‖ ⎤ ⎢ λ y ⎥ = T·⎢ λ⊥ ⎥ ⎢ ⎥ ⎢0⎥ ⎣ ⎦ ⎣ λ xy ⎦

(37)

1 + exp

with

ν′ =

hk λ⊥ k

To determine the effect of process-induced stress on the component level, the Young’s modulus E, thermal expansion coefficient α th and volumetric shrinkage due to the chemical reaction β V as a function of temperature and degree of cure have to be defined. By carrying out non-isothermal trials on the rheometer, the epoxy modulus Em as a function of degree of cure has been evaluated. At first, the gel point for different heating rates, which is defined by the intersection of loss- and storage modulus, was determined. As from the gel point, a 3D-network of the polymer is established and mechanical stresses can be transmitted. The degree of cure of this gel point was calculated in dependence of the temperature by using the developed reaction model for the specific heating rates, see Eq. (15) and Table 5. The average gel point is identified at a degree-of-cure of αGel = 0, 61 ± 0, 08. Analogue to the transition area of the specific heat capacity around the gelation point (see Eq. (26)), a similar approach is used to reproduce the progress of the modulus as a function of degree of cure [19].

(31)

λ‖ = λ f‖ φ + λ m (1 − φ)

n

Δε total (α, T ) = Δε el + Δε th + Δε chem.

Researches have shown a dependency between the thermal conductivity and temperature, respectively degree of cure but the impact on the curing process was estimated as insignificant [16,51]. For the manufacturing process, it is also assumed that no temperature gradient occurs on the external surfaces of the structure, so the only heat transfer exists in the through-thickness direction. Due to the considered thin structures, also a change of the thermal conductivity as a function of temperature or degree of cure would have a lower impact on the process so the thermal conductivity was presumed as constant [22].

λm = 0, 21 W/mK λ f‖ = λ f ⊥ = 1, 22 W/mK

htotal ∑k = 1

Thermal and chemical procedures during the curing lead to processinduced stresses which are based on occurring strains. They can be summarized as follows

∑k = 1 ρk cp, k hk ∑k = 1 ρk hk

(36)

3.3. Thermo-chemo-mechanical coupling

n

cptotal (α, T ) =

.

For the through-thickness-direction, a series connection of the single layers can be assumed, see Eq. (37) [22].

cp,f ρf φ + cp,m, t (α, T ) ρm (1 − φ) ρf φ + ρm (1 − φ)

total

k=1

Fig. 10. Evolution of specific heat capacity for a user defined temperature profile.

cp,UD (α, T ) =

∑

,

Table 5 Gel point αgel for different heating rates β .

(34)

with 6

heating rate β in K/min

gel point αgel

1 5 10 20

0,6 0,71 0,58 0,53

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Table 6 Identified parameters for calculating Em as a function of α . i=1

i=2

i=3

i=4

E (i) in GPa

0,059

0,23

0,065

/

a(i)

0,943

0,922

0,875

/

c (i)

0,007

0,013

0,022

0,873

Fig. 13. Evolution of the modulus for a user defined temperature profile.

Fig. 11. Comparison of measured and calculated modulus of the epoxy as a function of degree of cure.

Fig. 14. Comparison of measured and calculated thermal expansion as a function of temperature.

approach of Hashin/Hill [56,57]. A thermomechanical analysis (TMA) was used to measure the thermal expansion of the epoxy material. The positioned specimen was heated from 10 °C up to 190 ° C with a temperature ramp of 3 K/min , the elongation of the material depending on the temperature was logged and the coefficient of thermal expansion (CTE) α mth (T ) was calculated by using a secant method (see Fig. 14). The characteristic of the progress until the glass transition temperature can be approximated with a 3rd degree polynomial.

Fig. 12. Comparison of measured and calculated modulus of the epoxy as a function of temperature.

th α1,m (T )

A dynamic mechanical analysis (DMA) was performed to characterize the modulus of the fully cured epoxy material as a function of temperature, see Fig. 12. The mathematical description of this measurement was realized similar to the specific heat capacity due to the significant change of the material behavior around the glass transition temperature TG , see Eqs. (24)–(28). The linear decrease of the modulus in the beginning can be described with

E1,m (T ) = −0, 014 GPa/K·T + 3, 86 GPa.

= 5.9 × 10−11 1/K4·T 3 − 6.1 × 10−8 1/K3·T 2 + 2.1 × 10−5 1/K2·T − 2 × 10−3 1/K

Moreover, a linear increase of the thermal expansion coefficient (CTE) above TG was estimated with th α 2,m (T ) = 6.9 × 10−7 1/K2·T − 2.1 × 10−4 1/K

(41)

th α3,m (T ) =

E3,m (T ) =

1 + exp

(

TG1 − T cE

)

th th α 2,m (T ) − α1,m (T )

1 + exp

The combination of these two moduli, with respect to the change around the glass transition temperature can again be expressed with

E2,m (T ) − E1,m (T )

(44)

Again Eq. (45) was used to interlink the two approximations with respect to the transition area around TG1, whereas cα = 3, 34 was calculated.

(40)

For higher temperatures above the glass transition temperature, a stable modulus is assumed with

E2,m (T ) = 0, 23 GPa.

(43)

(

TG1 − T cα

)

th + α1,m (T )

(45)

The determination of the CTE as a function of degree of cure turned out to be not feasible. A TMA couldn’t be used due to the fact, that until αGel = 0, 61 ± 0, 08, no three-dimensional network of the polymer is established and therefore also no load can be transmitted. Above αGel the thermal expansion and chemical shrinkage superimpose and therefore distort the measurement [58]. In this case, a constant CTE above the gelation point was used, which was adopted from literature and showed good correlation with affiliated experiments [19]. So

+ E1,m (T ), (42)

the constant cE = 0, 87 was calculated with a least squares fit method. The connection between E (α ) and E (T ) is established by using a step function, equivalent to Eqs. (27) and (28). To illustrate the evolution of the modulus of the epoxy material, the dependency of this characteristic for a user-defined temperature profile is presented in Fig. 13. The modulus of the composite, which is influenced by the matrix and fiber modulus, was calculated with an

0, for α < αgel α mth (α ) = ⎧ . − 6 ⎨ 84.6 × 10 , for α > αgel ⎩ 7

(46)

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Therefore βm (α ) can be defined as

βm (α ) = 0, 019α.

(54)

To homogenize the matrix shrinkage onto the overall longitudinal shrinkage of the composite, an approach of Shapery was used [60,61].

β‖ =

(1 − φ) Em βm φEf‖ + (1 − φ) Em

(55)

The transversal chemical shrinkage was described with the following expression [57],

β⊥ = (1 − φ)(1 + νm ) βm − [ν13 φ + νm (1 − φ)], Fig. 15. Evolution of the CTE for a user defined temperature profile.

(1 − φ) E β

m ⎡ φE + (1 − φ)mE ⎤. m⎦ ⎣ f‖

As for the specific heat capacity and the modulus, also the link between dependency of degree of cure and temperature for the CTE is managed by a step function similar to Eqs. (27) and (28). The course of the CTE for the user defined temperature profile is shown in Fig. 15. The CTE for the matrix material is established, but the CTE for the unidirectional layer still has to be defined. For the transversal direction, an approach according to Dong was used which, from literature, has shown appropriate results compared with the experiments [21,17].

4. Numerical simulation and experiments Two different use cases were manufactured and subsequently simulated to evaluate the accuracy of the established procedure for predicting the curing kinetics, residual stresses and warpage on part level. At first, press molding trials of composite sheets were used to evaluate the accuracy of the reaction model. Furthermore metal-composite hybrid specimen were manufactured and the resulting deformation due to the different material characteristics were compared with the simulation. For the simulation, the tool Comsol Multiphysics [62] was used and the developed mathematical description of the cure kinetics and material properties were implemented into the software.

α⊥th= α mth (1 − νm ) + [α fth⊥ − α mth (1 − νm )] · 0, 96

0,2 Ef ⊥ 0,013 −0,036 1,24(Ef ⊥/ Em)−0,053νm νm φ Em

( )

(47)

The Poisson’s ration of the matrix material νm and the fiber properties were taken from literature [21]. The longitudinal CTE of the composite was calculated with the following empirical approach [22].

α‖th =

α f‖th Ef‖ φ

4.1. Press molding of composite sheet

α mth Em (1

+ − φ) Ef‖ φ + Em (1 − φ)

The composite sheet with a layup of [0, 90]sym was press molded at 180 ° C by using conventional press technique. Two material temperatures were recorded. A temperature sensor was located in the middle of the sheet between layer 2 and 3 and a near-surface temperature sensor was logging the surface temperature of the composite sheet. For the simulation, this surface temperature was used as a boundary condition. The temperature profile of the sensor, located in between the layers, was used to evaluate the accuracy of the simulated temperature profile. Also an ultrasonic sensor was integrated into the tooling to measure online the degree of cure. The results of the manufacturing trials in comparison with the simulated temperature profile of the middle sensor (Fig. 16a) and the simulated degree of cure (Fig. 16b) show a good accordance and therefore confirm the findings regarding the reaction model and thermo-chemical coupling.

(48)

The transformation of the CTE from local to global coordinate system can be performed similar to Eqs. (34) and (35). Because of the specific material characteristic of the matrix material, the following assumption can also be defined. For α < αGel

α th‖ = α thf ‖, α th⊥ = 0.

(49)

Finally the chemical shrinkage will be evaluated, which describes the volumetric shrinkage of the matrix material due to the progressing polymerization. As mentioned before, the determination of chemical shrinkage and thermal expansion cannot be separated due to the fast curing abilities of the matrix material. Preceded researches have shown a direct and linear dependency between degree of cure and chemical shrinkage [49,59]. Assuming a uniform shrinkage of the matrix material, which is independent of the direction, the chemical shrinkage Δεmch can be defined as follows [57].

Δεmch

=

3

1 + ΔVm − 1

4.2. Manufacturing of metal-composite hybrid specimen The manufactured specimen had a size of 200 mm × 25 mm and a layup of [0]4 directly and entirely applied on the steel specimen. Because of the different material characteristics of metal and composite, internal stresses occur during the manufacturing process and lead to warpage. This warpage ought to be measured by an optical system

(50)

with

ΔVm = β V ·Δα

(51)

ΔVm defines the specific change of the volume while curing and β V the volumetric shrinkage coefficient. A simplified linearization of Eq. (50) leads to Δεmch =

3

1 + ΔVm − 1,

= [ 3 1 + β V − 1]Δα, = βm Δα.

(52)

The results of the density measurement were used to define β V .

βV =

ρm (α = 1) − ρm (α = 0) = 5, 9 % ρm (α = 0)

(56)

Fig. 16. Evaluation of reaction kinetics approach.

(53) 8

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Fig. 17. Test rig for measuring the warpage of metal-composite hybrid specimen while the curing and cooling process. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 19. Comparison of temperature profiles for a temperature and degree of cure dependent approach and an approach with constant parameters.

necessity of temperature und degree of cure dependent characteristic values of the matrix material on the accuracy of the process simulation. The cure kinetics themselves were implemented as described in Section 3.1 and thus not variated. Both use cases, the press molding of a composite sheet and metal-composite hybrid specimen were examined.

while the curing and cooling process to validate the process simulation, so a press-molding process was not feasible due to the rigid press tool. For this reason, a test rig was built, which is shown in Fig. 17. The non-cured metal-composite hybrid specimen was fixed laterally on the carriage and pulled into the heated convection oven. After the curing, the oven was opened and so the specimen cooled down again. The emerging warpage was recorded with an optical measuring system. Temperature sensors on the specimen logged the temperature, this temperature profile was used as a boundary condition for the following process simulation. To reduce the computing time and complexity of the simulation model, also the degree of cure was calculated by using the logged temperature profile and therefore provided as a boundary condition for the process simulation. So a self-heating in between the layers due to the exothermal reaction was not considered. But because of the sluggish heating of the specimen and consequently slow curing in combination with the thin composite build-up, an extend of additional heat generation is not to be expected. Fig. 18a and b show the results of the manufacturing trials and simulation. The graphs show a good correlation between simulation and experiments. After approx. 4 min, the gelation point is passed and the matrix can transmit first loads. Until reaching the maximum degree of cure, a positive deflection of 0,12 mm was determined which is a result of chemical and thermal effects. In the beginning of the cooling process, no significant change of the deflection was recognized because of the low modulus of the epoxy above the glass transition temperature. Below this glass transition temperature, the modulus increases and lead to a negative deflection down to −2,3 mm.

5.1. Press molding of composite sheet For this use case, the matrix density ρm (α ) and specific heat capacity of the matrix cp,m, t (α, T ) are the characteristic values of interest. For evaluating the influence of the degree-of-cure and/or temperature on the prediction accuracy, these values are set alternating to their constant values at ambient temperature respectively full cure. Two different deviations were analyzed regarding the peak cure temperature, the variance of maximum temperature Tmax (based on exothermal effects) and the variance of time offset of this peak toffset . Fig. 19 clarifies this procedure and shows the difference between the temperature – and degree-of-cure-depending approach and an approach with constant parameters. The results of this sensitivity analysis for different parameter combinations are shown in Table 7. The results reveal a negligible dependency of the density on the prediction accuracy of the simulation. Also a minor impact shows the temperature-dependency of the specific heat capacity on the results. In contrary, the specific heat capacity as a function of degree of cure has a major influence on the prediction accuracy. If cp,m isn’t described taking into account the degree-of-cure dependency, a deviation up to 8 % regarding the temperature peak and 45 % regarding the time shift was examined. These results are explainable with the characteristic of the specific heat capacity as a function of degree of cure (see Fig. 8). In the beginning of the press molding, the prepreg has a temperature of 23 °C and a cp,m (α = 0) = 4, 34 kJ/kg K . In the press mold, the material is heated above 180 ° C and starts to cure around 100 °C, the specific heat capacity drops linearly to cp,m (α = 1) = 2, 22 kJ/kg K . Consequently, for an approach with a degree-of-cure dependency of the specific heat capacity, more energy is needed to heat up the material until fully

5. Sensitivity study The present research has shown a good accordance between simulation and trials, but the characterization of the matrix material as a function of temperature and degree of cure was very complex and timeand cost consuming. Preferably this complexity should be reduced for future matrix materials to accelerate upcoming research projects. Therefore a sensitivity study was realized with the aim to identify the

Table 7 Press-molding of a composite sheet: Impact of temperature- and degree-of-cure depend values on the process simulation compared with the experiment.

Fig. 18. Experiment and simulation of hybrid specimen. 9

ΔTmax in %

Δtoffset in %

cp,m, t (α, T );ρm (α )

2,98

0,84

cp,m, t (α );ρm (α )

3,05

0,98

cp,m, t (T );ρm (α )

40,27

7,44

cp,m, t = const. ;ρm (α )

41,96

6,36

cp,m, t (α, T );ρm = const.

2,98

0,94

cp,m, t (α );ρm = const.

2,95

0,95

cp,m, t (T );ρm = const.

41,01

4,17

cp,m, t = const. ;ρm = const.

41,01

4,37

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constant values, a discrepancy of deflection of 2, 73 mm regarding the experimental results is recognized and clarifies the need for dependent characteristic values. 6. Conclusion This paper has focused on the characterization and mathematical description of the cure kinetics, the thermo-chemical and thermochemo-mechanical behavior of an epoxy matrix material, respectively of a pre-impregnated continuous glass fiber reinforced epoxy prepreg. Following, a process simulation to predict the curing of a press molding process and the warpage of a metal-composite hybrid specimen was performed and compared with results of manufacturing trials. Generally a very good prediction accuracy was determined when implementing the introduced characteristic value as a function of degreeof-cure and temperature into the simulation. Furthermore a sensitivity analysis has shown the necessity of temperature and degree-of-cure dependent characteristic values on the accuracy of the simulation. For the press molding of the composite sheet, it was shown that the specific heat capacity of the matrix material should be characterized entirely to predict the temperature profile of the material and subsequently the curing itself with an acceptable accuracy. For the simulation of the metal-composite hybrid specimen it was shown, that the CTE and their determination has a major influence on the result. Also a non-negligible influence on the accuracy was identified in the temperature and degreeof-cure dependency of the modulus of the epoxy material, although lesser than the CTE. The impact of the chemical shrinkage on the result was measurable but insignificant. Still, these studies have been performed on part level with a low complexity. Especially the metal-composite hybrid specimen had shown a considerable deflection due to the minimal geometrical moment of inertia, so the upcoming residual stress was mainly transformed into deformation. Further investigation will focus on the influence of this manufacturing process on the deformation and internal stress of more complex geometries to transfer the achieved knowledge into upcoming part developments.

Fig. 20. Comparison of the warpage for a temperature and degree of cure dependent approach and an approach with constant parameters. Table 8 Manufacturing of a metal-composite hybrid specimen: Impact of temperatureand degree-of-cure depend values on the process simulation compared with the experiment. Δscuring

Δstotal

in mm

in mm

th Em (α, T );α m (α, T );βm (α )

0,02

0,06

th Em (α, T );α m = const. ;βm (α )

3,01

2,84

th Em (T );α m (α, T );βm (α )

0,03

0,08

th Em (α );α m (α, T );βm (α )

0,04

0,08

0,71

0,61

2,85

2,68

3,64

2,73

th Em = const. ;α m (α, T );βm (α ) th Em (α, T );α m (α, T );βm = 0 th Em = const. ;α m = const. ;βm =

0

cured. In contrary to an approach with a constant cp,m , characterized at full cure, the curing time extends and peak temperature decreases. So it can be noted, that a characterization of the specific heat capacity of the matrix material depending on the degree-of-cure is necessary for a precise cure process simulation on part level, the other dependencies are not crucial.

Author contributions Robert Thomas: Conceptualization, data curation, Investigation, Methology, Validation, Writing original draft, writing review and editing. Simon Wehler: Investigation, Methodology, validation. Maik Gude: Supervision. Fabian Fischer: Supervision, Project administration. Basically, Simon and I have done the research, Fabian is our boss and helped from the "economical" site and gave us the necessary time to do all the experiments Prof. Maik Gude is my PhD supervisor.

5.2. Manufacturing of metal-composite hybrid specimen For this use case, the impact of the following temperature- and degree-of-cure dependent characteristic values on the accuracy of the simulation have been evaluated: Em (α, T ), α mth (α, T ) and βm (α ) . It was shown, that the mechanical performance of the material changes during the process, following the influence on the example of a metal-composite hybrid specimen will be examined. Again two different deviations were examined. The discrepancy of trial and simulation regarding the warpage after the heating- and curing process scuring and the entire warpage after cooling down to room temperature stotal (see Fig. 20). A selection of significant results of the sensitivity analysis is shown in Table 8. A simulation with full temperature and degree-of-cure dependency has shown an insignificant discrepancy of 0, 06 mm and so confirms the outlined approach of simulating the warpage of the metal-composite hybrid specimen. A major impact is seen in the temperature- and degree-of-cure dependency of the CTE. This value has shown the highest impact on the accuracy of the simulation, so the thermal expansion coefficient should always be characterized as a function of temperature and degree of cure. For the modulus and solely a temperature or degreeof-cure dependency, the prediction accuracy is still reasonable, but a constant modulus shows a higher deviation of 0, 61 mm . Also, the sensitivity study has revealed, that the influence of the chemical shrinkage on the result is minimal. The first load transmission occurs after the gelation point and a degree-of-cure of α = 0, 61, so most of the resulting stress directly dissolves. By simulating this use case with

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