Visco-thermo-elastic Simulation Approach For Prediction of Cure-induced Residual Stresses in Fiber Reinforced Composites

Jonas Müller. Institute for Plastics Processing, RWTH Aachen, Seffenter Weg 201, 52074 Aachen. Michael Müller-Pabel. Institute of Lightweight Engineering and Polymer Technology, TU Dresden, Holbeinstraße 3, 01307 Dresden. Corresponding author: Michael Müller-Pabel. E-mail address: michael.mueller-pabel@tu-dresden.de Niklas Lorenz. Institute for Plastics Processing, RWTH Aachen, Seffenter Weg 201, 52074 Aachen. Benjamin Gröger. Institute of Lightweight Engineering and Polymer Technology, TU Dresden, Holbeinstraße 3, 01307 Dresden. Johannes Gerritzen. Institute of Lightweight Engineering and Polymer Technology, TU Dresden, Holbeinstraße 3, 01307 Dresden. Maik Gude. Institute of Lightweight Engineering and Polymer Technology, TU Dresden, Holbeinstraße 3, 01307 Dresden. Christian Hopmann. Institute for Plastics Processing, RWTH Aachen, Seffenter Weg 201, 52074 Aachen.


Intr Introduction oduction
The future use of fiber reinforced polymers (FRP) for series production of lightweight structures primarily depends on availability of competitive, cost-efficient and reproducible manufacturing technologies. Liquid composite molding (LCM) represents a well-established production route, which nevertheless suffers from certain drawbacks associated with the complex FRP material behavior. Especially the interaction of fiber and matrix as well as the cure-dependent resin properties represent major challenges for part design and process control. As the thermo-mechanical behavior of the resin depends on a number of process parameters like time, temperature, degree of cure and pressure, material characterization and modelling require a comprehensive strategy.
Previously published work on cure-dependent resin behavior mainly focused on better understanding of conventional prepreg and RTM manufacturing processes [1][2][3][4]. Only few attempts were made to analyze the process-dependent properties of fast curing resins [5,6]. Furthermore, pressure dependence is often neglected, which is questionable when it comes to newly developed technologies with high impregnation speed like high pressure RTM (HP- RTM) or wet compression molding [7]. To fill this gap, an appropriate material model approach in combination with reliable material data are required. For this purpose, a visco-thermo-elastic simulation approach is selected [8] and calibrated. Cure-dependent properties were determined by shear rheological and dynamic-mechanical analyses (DMA) on partially cured neat resin specimens. A newly developed volumetric dilatometer is used to analyze pressure-und cure-dependent chemical shrinkage at process-related conditions [9]. The determined relationships between resin behavior and process parameters are transferred to the material model and used for simulations on micro-scale aiming at reproduction of surface waviness effects and residual stresses. The experimental program described here is complementary to a previously published work of the authors [10]. In this contribution, special attention is paid to pressure-dependent chemical shrinkage and cure-dependence of viscothermo-elastic modulus of the fast curing resin system TRAC Epikote 06150, which is used for RTM processes in automotive industry. Isothermal (80, 90, 100 and 120°C) measurements at different pressure loads (5, 10, 25 and 60 bar) are carried out using a dilatometer optimized for the characterization of fast curing resin systems [2,9].
Samples from the cured dilatometer specimens are prepared for the subsequent DSC measurement. The DSC analysis is conducted using the Q2000 device from TA Instruments Inc., Austin, USA, according to the specifications described in DIN 65467. A heating and cooling rate of 10 K/min is selected for all measurements and applied to a temperature range from -50°C to 250°C. The degree of cure (DOC) is determined from the value of remaining reaction enthalpy ∆ and the total reaction enthalpy ∆ , according to eq. 1 [11]: The cure-dependent visco-thermo-elastic properties were determined using an Anton Paar MCR 502 rheometer device.
A plateplate measurement system with a diameter of 25 mm and a gap of 1.5 mm was used to measure in the liquid stage as well as the liquid-solid transition during gelation, which was determined to occur at = 0.68. To allow continuous measurements, oscillation mode at a frequency of 1 Hz and a shear deformation amplitude of 1 % were chosen. Isothermal temperatures of 60°C, 80°C, 100°C and 120°C were used. The data are transferred from time scale to DOC scale by using the previously published reaction kinetics model of the resin system [10].
In order to cover the entire post-gelation area, a solid rectangular fixture (SRF) was used to analyze the influence of varying DOCs using specimens with a cross section of 2 mm x 4 mm and a length of 40 mm. In order to obtain time-dependent modulus data, relaxation experiments with an initial shear deformation of 0.1 % were performed at 5 K temperature intervals ranging from room temperature to a temperature 10 K below the actual glass transition.
A relaxation time of 5 min was required to generate overlapping stiffness values between the selected temperature segments. Partial cure was realized by using a pre-cure schedule of 7 h at 60°C within a molding tool; followed by progressive post-cure schedules according to Table 1 that were performed in a convection oven. Thermal expansion and chemical shrinkage are implemented by using UEXPAN.
In order to calculate the cure state , a n th -order and an autocatalytic type was selected (eq. 2 and 3): with Ai, ni, m, Ei being material parameters and the universal gas constant defined in [10] with 0 , g∞, being material parameters determined by DSC scans on partially cured resin material and linear regression. Values are listed in Table 2.
T During curing of the resin, mechanical properties are steadily changing due to network growth, gelation and vitrification. To model the cure-dependent relaxation curves extracted from experiments, in a first step Prony series is fitted to generated master curves for ξ = 1 according to eq. 5 [10]. matches the ℎ experimental master curve, while = 1 corresponds to the master curve at ξ = 1: is determined by fit with lowest residual mean square error (RMSE). Generally, 1 to 15 Prony elements are tested.
According to O'Brien et al. [15] equilibrium shear modulus ∞ can be calculated by eq. 6: where Cg∞, Dg∞ and Fg∞ are material parameters and represents the degree of cure at gelation.
For fitting the remaining master curves, following workflow is used: • Hence relaxation data at lower is unknown, data for high relaxation times is augmented following eq. 6 for > 10 13 -Prony series is fitted with equal to for = 1 • Equilibrium shear stiffness is used for calculation of ( , ).
Based on these Prony fits, (t) for ξ k+1 ≤ ξ ≤ ξ k is interpolated. For ξ < ξ max( ) eq. 8 is used to calculate time-dependent shear modulus: Temperature-and cure-dependence are considered by shift factor which is calculated by a modified WLF-approach [13]: Visco-thermo-elastic Simulation Approach For Prediction of Cure-induced Residual Stress...

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1, 2 are experimentally determined parameters. 0 is the selected reference temperature of 40°C. is substituted by reduced time following eq. 10: Due to the use of UEXPAN for thermal expansion and chemical shrinkage, providing thermal strain increment Δ ℎ within UEXPAN is necessary. Generally chemical volume change ℎ is expressed by chemical strain increment Δ ℎ assuming a cube with side length as followed: Assuming that the volumetric shrinkage is proportional to at a constant pressure leads to: where ( ) is an experimentally determined function (cf. eq. 17).
Following this, Δ ℎ is equal to the sum of actual thermal expansion increment Δ ℎ and chemical shrinkage increment ℎ: Δ ℎ is calculated as followed, where thermal expansion coefficient ℎ is assumed as constant defined in [ The results of volumetric dilatometer and DSC measurements are depicted in Figure 1. According to previous studies a linear approach was chosen to model the development of the chemical shrinkage after gelation [2,16]. For gelation time and corresponding DOC values the mean value of = 0.68 was used according to [10]. Considering eq. 14 the pressure dependence of the chemical volume shrinkage can be fitted by eq. 17 and the data contained in Figure 1. For high pressure load eq. 17 approaches a limiting value.
With 1, 2, 3 representing fitting parameters and values listed in Table 3. proved by means of molecular dynamic simulations that the free volume fraction increases during the curing process [18]. As the free volume fraction reduces with high pressure loads an increasing change of the volume can be expected with elevated curing pressures [19].
In order to provide experimental reference data for definition of the Prony parameters, cure-dependent data for the instantaneous modulus 0 and the equilibrium modulus ∞ as well as the relaxation behavior between glassy and rubbery regime are required. While the determination of 0 can be realized straightforward by evaluating the initial stress response of the relaxation experiments at low temperatures, the determination of ∞ is more challenging due to possible post-cure when using partially cross-linked specimens. For this reason, values for ∞ at < 1 are taken from the isothermal oscillating measurements in plate-plate mode. The evaluation of these experiments is limited to a DOC of 0.85, as in plate-plate-mode the increasing resin stiffness leads to residual stresses in combination with hindered lateral contraction. Consequently, a relaxation experiment on a fully cured specimen is used to generate an additional value for ∞ at = 1. The results are shown in Figure 2 together with the fit based on the work of [20] and according to eq. 6. Exact values for eq. 6 are listed in Table 4.   The cure-and time-dependent relaxation behavior of the resin system is evaluated by using the time-temperature superposition principle. Individual master curves are constructed for the DOC from 0.88 to 1 at a reference temperature of 40°C. The results with corresponding Prony series fit with = 13 following eq. 5 are shown in Figure 3. During fitting a RMSE of 0.001 was achieved. The derived Prony series are in good agreement with the experimental data.
When plotting the master curves on the reduced time scale, it can be seen that the relaxation time increases with cure and the instantaneous modulus 0 decreases with cure, which can be attributed to reduced mobility during cooling at higher degrees of cure [21]. Figure 4 shows shift factors and the corresponding fits following eq. 9. Generally, the slope of time-temperature shift factors log ( ) decrease with cure. Furthermore, to obtain good fitting results the parameters 1 and 2 are fitted individually. The individual Parameters can be found in Table 5. Comparison between shift f een shift fact actors fr ors from mast om master curv er curve gener e generation and function fitt ation and function fitted (cf. eq. 9) ed (cf. eq. 9) T Table 5. P able 5. Par aramet ameters of eq. 9. ers of eq. 9.

Simulation r Simulation results esults
In a first step, a verification of the suggested approach was performed by using the unit length cube model. A set of simulations were executed in order to simulate time dependence of shear modulus. Therefore, simulations at different temperatures between 20°C and 130°C in 5 K steps and varying = were performed. Figure 5 shows exemplarily depicted results for = 1.0 in comparison to input data of the experimental relaxation tests. It shows that simulation data is in good agreement with experimental data and the material behavior is reproduced correctly.
Figur Figure 5: Comparison betw e 5: Comparison between input data and simulation data een input data and simulation data In a second step, the material model presented was used to compute the displacement of the outer surface of a RVE after a generic cure cycle. The generic cure cycle consists of 4 phases (cf. Figure 6). First, the model is heated up from 60°C to 120°C with 20 K/min. In the second step, the model is subjected to a constant temperature of 120°C for 20 min in order to completely cure the resin. Afterwards, it is cooled down to 25°C with -20 K/min. For a fixed top layer of the RVE the boundary is deleted which represents demolding, and the model is subjected to another 2 min at 25°C. For simulation in case of a free surface, surface is hindered in moving in positive 2-direction. Figure 6 shows This results mainly in additional contraction of the material due to chemical shrinkage and a different relaxation behavior at lower ξ.
The results for a fixed top layer differ only slightly, due to fixed surface and high relaxation at higher temperatures of residual stresses mainly induced by chemical shrinkage. Comparing the results to a free surface shows huge differences.
This is based on chemical shrinkage due to unhindered deformation (cf. Figure 6). The production of composite materials using fast curing resin systems has gained considerable interest in recent years. However, further advances in quality and reproducibility are required, especially for automotive applications.
Component distortion and surface defects resulting from chemical shrinkage and thermal expansion pose major challenges that require costly quality control and process monitoring. The presented work aims at a comprehensive characterization and modeling of the underlying phenomena that lead to surface waviness and part distortions.
In this work, the material characterization of a fast curing epoxy resin based on [10] was extended by cure-dependent mechanical properties and pressure-dependent chemical shrinkage. The experimentally determined results were transferred into a material model to enable numerical studies of surface waviness and part distortion induced by manufacturing process. A RVE modeling method was proposed that allows sensitivity analysis.
In terms of verification, the developed cure-dependent viscoelastic material model shows good agreement with the relaxation test data. Regarding the selected RVE model, the visoc-elastic matrix behavior was shown to have only limited sensitivity to a change of surface the boundary condition. This does not apply to the cure-dependent viscothermo-elastic matrix behavior. In this case, the boundary condition during processing has a considerable influence on the resulting surface waviness. The assumption of a perfectly fixed top layer during cooling results in a significantly lower surface deformation compared to a free surface since deformations resulting from chemical shrinkage where hindered by the fixed top layer. Overall, the cure-dependent visco-thermo-elastic matrix behavior yields a maximum of deformation for an unconstrained surface.
Future work will focus on verification of the made assumptions and on modelling a multi-scale approach in order to