1. Introduction
Fiber-reinforced polymers (FRPs) are commonly used in aerospace and high-performance applications due to their higher strength-to-weight ratio. They are made with a fiber reinforcement embedded in a polymeric matrix. This reinforcement ranges from short fibers, continuous long fibers and fabrics. The most common fiber reinforcement materials are glass fibers (GFRPs), carbon fibers (CFRPs) and other organic fibers such as Kevlar, Dyneema, etc. [
1]. Composite materials possess some advantages related to traditional (metallic) materials in addition to the weight reduction, including toughening for impact, fatigue resistance, corrosion resistance, electromagnetic transparency, erosion and wear resistance, acoustic and vibration damping, low thermal expansion, among others [
1,
2,
3,
4]. However, the most important advantage of FRPs is their tailoring ability to fit design requirements.
Two families of polymeric matrices are commonly used in FRPs, namely thermoplastics and thermosets. While thermoplastics are processed by melting the polymer, and cooling it down to its final configuration, thermoset-based composites are manufactured with a non-reversible thermo-curing process where the polymer is transformed from a gel to a monolithic material. The latter have been the preferred option in aerospace applications due to their easier processing characteristics.
Inadequate manufacturing processes in composite materials generate low-quality products and performance reduction in as-manufactured conditions due to residual stresses, voids formation, or incomplete curing. This performance reduction can lead to design failure because material properties were overestimated, or because the geometry does not fit the requirements [
5,
6]. Linking of composites manufacturing simulation with in-service performance is a recent research topic that has become affordable due to increased computing power, but we still lack a complete understanding of how it influences the final performance.
Focusing on carbon fiber-based thermoset composite systems (CFRPs), micro-residual stresses appear at the microstructural level due to the mismatch between the thermal expansion coefficients of the fiber and the polymeric matrix, and due to the chemical shrinkage, that takes place during the curing process [
7,
8,
9,
10,
11]. Additional residual stresses can be obtained in composite laminates due to the anisotropy of the plies, but they are out of the scope of this work, which will focus on the micro- (constituents) scale of polymer matrix composites.
The curing process is inherently a thermo-chemical phenomenon for thermoset polymers where an exothermic chemical reaction starts at a given temperature and finishes when the polymerization process ends. One of the most commonly used approximation models for polymer curing was proposed by Kamal [
12]. This model relates the curing state in the material with the time and temperature evolution. Currently, several works can be found in the literature regarding the experimental characterization of the curing behavior of polymeric resins, including epoxies [
13,
14,
15,
16,
17,
18,
19,
20]. Many of them use differential scanning calorimetry (DSC) and dynamical mechanical analysis (DMA) to identify the curing behavior and the evolution of the elastic modulus.
Experimental works to measure actual residual stresses at micro-scale level are not easy to perform due to the reduced length scale at which they occur. For instance, Minakushi [
21] performs fiber-optic based tests to measure cure shrinkage in fiber-reinforced laminates, obtaining local strain measurements in the fiber direction and through the thickness, while Seers et al. [
22] presents a recent review of the measurement techniques of residual stresses in composites, most of them relying on macro-measurements to obtain indirect measures of the micro-residual stresses. Similar difficulties arise regarding the characterization of the mechanical properties of the material at micro-scale level due to size effects [
23], especially for the fracture toughness.
Numerical simulation of the properties evolution with curing is a current research field. To the best of the authors’ knowledge, Ersoy et al. [
8] is among the first studies to deal with these simulations, considering only the elastic part of the material. Yuan et al. [
24,
25] uses viscoelastic constitutive modeling for the elastic modulus evolution during curing and representative volume elements (RVEs) consisting of one fiber or multiple fibers coupled with a multi-scale approach to capture temperature profiles from the macro-scale.
A similar approach was followed by Hui et al. [
26], who also use a multi-scale approach to analyze the heat transfer effects of the curing process at the micro-scale level. They include an extended Drucker–Prager model to account for the polymer failure response. Danzi et al. [
10] instead used a more advanced material constitutive model that accounts for the plastic response and failure of the polymer matrix, a model that was previously proposed by Melro et al. [
27] and has been used by several authors to simulate the mechanical response of epoxies. All these works converge in the usage of RVE models [
28,
29] to analyze the mechanical response of the composite material at the micro-scale level with the appropriate boundary conditions (BCs) to guarantee solution consistency [
30].
Another relevant topic relates to the methodologies used to estimate the effective macro-scale mechanical properties and the imposed BCs. Volumetric average measures of stress and strain fields are the most commonly used technique to extract the effective strain and stress response [
28]. Second order homogenization theories are also available in the literature [
31] but are not considered in this work because they require the usage of strain gradient measures, which falls outside the scope of this work.
Considering first order approximations, several authors have studied the influence of the BCs in the effective strain–stress response, concluding that the periodic boundary conditions (PBCs) give the best compromise between accuracy and computational simplicity [
32,
33]. However, the usage of PBCs has limitations in the simulation of strain localization phenomena such as damage propagation, because they over-constrain the strain field at the boundaries [
28,
32,
34]. Nevertheless, PBCs provide accurate estimates up to failure initiation, which is the focus of this work.
Here, a micromechanical model is developed using the commercial software Abaqus® and user-material (UMAT) subroutines to simulate the curing process and estimate the micro-residual stresses and its influence on the mechanical performance of an aerospace grade epoxy resin. A distinction between residual (macro-scale) and micro-residual (micro-scale) stresses is made because the micro-residual stresses are developed inside the RVE with traction-free boundary conditions.
Using the methodology proposed by Danzi et al. [
10] that couples the polymer constitutive formulation proposed by Melro et al. [
27] with the curing kinetics model, an enhanced temperature dependence of the material properties, especially for the elastic properties, is included in this work to appropriately model the shrinkage and the cooling processes.
The effective strain and stress measures are extracted directly from the PBCs output, i.e., the strain is obtained from the strain of the master node, and the stress from the force-conjugate measure of the applied strain, instead of using volumetric averages that are not appropriate when strain localization is achieved. The predictions obtained are then compared with experimental data obtained from standardized composite macro-scale tests.
A detailed analysis of the RVE response during the curing process is also performed using three different RVE geometries. Instantaneous measures of the effective macro-strain are extracted to see effective material expansion and contraction, besides the typical stress analysis. This analysis shows how the constituents are subjected to local stresses while keeping the macro effective stress equal to zero.
Additionally, due to the uncertainties of the material parameters at micro-scale level, a sensitivity analysis is performed in this work, to assess the influence of the thermal expansion coefficient, chemical shrinkage, resin elastic modulus and cure temperature on the residual stresses and the posterior effective mechanical properties, to complement the overall analysis of the micro-residual stresses.
The following section describes the materials, the constitutive modeling approach, the finite element procedures and the micro-residual stress analysis methodology. Then, the results are analyzed to highlight the micro-residual stresses distribution inside the RVE, and to link their influence on the effective mechanical properties. Finally, conclusions are drawn based on the discussion of the principal findings of this study.
3. Results and Discussion
3.1. Experimental Results
The curing model parameters for the epoxy resin used in this study were obtained by fitting the experimental data of the curing process measured with DSC tests. The parameters are given in
Table 6.
Figure 4a shows the results of one representative DMA curing test. It shows how the elastic modulus begins to grow after the temperature of 100 °C is reached while the curing level develops.
Using the maximum obtained storage modulus and Equation (7) to obtain the instantaneous values of the shift function
,
Figure 4b shows the evolution of the shift function
and the curing state of the material with time. It shows the storage modulus follows the curing state response with some delay.
Accordingly, the shift function given in Equation (7) is fitted, obtaining the parameters
β = 4 and
γ = 0.8. A comparison between the experimental relation and the relation from Equation (7) is given in
Figure 5a.
Figure 5b shows the estimated mechanical properties of the resin system using Equations (7)–(9), which will be implemented in the numerical material models.
The tests coupons after mechanical testing are shown in
Figure 6. Two transverse tensile coupons failed near the tabs; however, the measured strength has a low deviation, and the values are similar to the ones reported by the manufacturer and other test campaigns, making them suitable for the scope of this work. The failure modes for the longitudinal and transverse shear tests are within the expected behavior.
A summary of the experimental results is given in
Table 7.
3.2. Residual Stress Analysis
During the curing process, the elastic modulus is developed at the same time the material expands due to the effects of the thermal expansion, and contracts due to chemical shrinkage. If the matrix material is completely free to deform it is not possible to generate stresses at the micro-mechanical level. However, in the case of fiber-reinforced composites, the fibers introduce local restrictions to the free deformation of the resin leading to the generation of local micro-residual stresses, the first stress generating mechanism. The second stress generating mechanism is the CTE incompatibility between the two constituents that becomes more relevant during the cooling-down stage.
To analyze the generated micro-residual stresses,
Figure 7 and
Figure 8 show the stress distribution for the three different RVEs by means of the von Mises stress and the pressure stress. These measures were selected because they correspond the stress invariants that define the yielding and the failure criteria given in Equations (11) and (18).
The regions between the fibers concentrate higher deviatoric (von Mises) stresses in the matrix, being the most favorable regions to damage initiation and propagation. On the other hand, the hydrostatic stress distribution in the matrix is predominantly negative, which means that the bulk is in tensile state (remembering that ), while the fibers remain in compression, as expected since the whole RVE is in global equilibrium and under traction-free boundary conditions.
To investigate the stress components evolution during the curing process,
Figure 9 shows the von Mises stress and the hydrostatic stress components evolution as a function of time, and
Figure 10 shows the effective RVE strain measured at the master nodes. Temperature is also superposed to ease the analysis. In this analysis, the stress measures correspond to a volumetric average over the matrix volume region. During the heating stage prior to the initiation of the curing in the polymer, no stress is generated and the RVE expands due to the combined effect of the positive CTE of the fibers and resin in the transverse direction, while it contracts in the longitudinal direction due to the negative CTE of the fibers.
Next, the curing process begins, and the matrix shrinks at a constant temperature (inside the plateau), until the curing process stops, achieving a stress equilibrium state with a smaller volume. A cooling stage follows, where additional straining is retrieved, developing more micro-residual stresses.
The strain measure evolution with time and the degree of cure present trends similar to the strain measured experimentally with optical fibers by Minakushi [
21].
Therefore, the matrix volumetric shrinkage is the precursor of the stress development during curing because it occurs at constant temperature, while the thermal contraction (CTE) is the stress precursor during cooling. The longitudinal strain deformation of the RVE is very low because it is dominated by the high stiffness of the fibers and their low negative CTE.
The chemical shrinkage for this resin is 2%, while the thermal contraction during cooling is only 0.6% (CTE multiplied by the temperature amplitude—from curing temperature to room temperature). However, the final volume reduction is around 0.9% estimated from the final volume of the RVE as presented
Figure 11.
If a full restrained analysis is performed, the volume contraction is enough to cause material failure, because the failure strain is lower than 0.7%. For the current example, no damage is retrieved from the numerical results, and only small plastic straining is found in the stress concentration zones.
Other results that can be extracted from the curing analysis are the failure index measurements in the RVE given by Equations (11) and (18). For this purpose,
Figure 12 shows the maximum failure index retrieved from the RVE as a function of time. It is clear from the previous figures that the micro-residual stresses achieve representative values especially between the narrowest fiber gaps and these stresses lead to the development of plastic strains. For the current example, no damage is achieved but the failure index is above 0.9.
3.3. Effective Mechanical Properties
To study the influence of the micro-residual stresses in the mechanical performance of the composite material, several simulations were performed using the selected RVEs subjected to different loading conditions as planned in
Table 3, giving special attention to the properties dominated by the matrix behavior. Two sets of simulations are analyzed: the first corresponds to an analysis with the nominal material properties without considering the curing analysis and without the micro-residual stress effects, which is named “no-cure”, and another set that is analyzed using the micro-residual stresses and the actual material condition coming from the curing step analysis, named “cure”.
The stress–strain response for the transverse tensile loading case is presented in
Figure 13. A direct comparison between the three different RVEs for the “no-cure” and “cure” condition is made. First, the overall response of the RVE is linear until failure, although the matrix resin is subjected to some plastic deformation. The elastic modulus remains unaffected because the material is completely cured, but the transverse tensile strength is reduced by the micro-residual stresses as expected.
A similar response is retrieved for the shear loading simulations in both transverse and longitudinal directions, as shown in
Figure 14. The shear modulus is practically unaffected by the micro-residual stresses and only the shear strength is reduced.
Figure 15 shows the fracture pattern for each loading case for the “no-cure” and “cure” tests. A similar failure pattern is retrieved for both cases because it is given by the loading condition, with the only difference in the initiation point that changes the position of the crack bands. In the case of the cure analysis, the initiation point is influenced by the micro-residual stress levels, unlike the no-cure analysis.
To facilitate the comparison of results between both analysis sets,
Table 8 shows a summary of the predicted effective mechanical properties and the experimental results. From the tensile tests, the transverse elastic modulus
and the Poisson’s ratio
from the experimental measurements are similar to the numerical predictions, the transverse tensile strength
presents a reduction of 11% from the “no-cure” condition although the experimental measured value is slightly lower. This difference can be attributed to the effect of the fiber-matrix interface and other manufacturing defects which are not considered in the current analysis [
28,
42]. Another relevant result that should be highlighted is the fact that the tensile strength of the composite is almost 30% lower than the bulk matrix strength, a tendency that is commonly highlighted in the literature [
9,
10].
A similar trend is observed for the predicted transverse and longitudinal shear moduli, which are similar to the experimental results, while the strength measures present a reduction of 22% and 12%, respectively, when compared with the “no-cure” condition. Similarly, the experimental measured values are lower than the numerical predictions. The effective properties estimated considering the cure condition are closer to the experimental results.
These results show evidence that the micro-residual stresses in the RVE after the curing process can reduce the material strength and should be considered when performing homogenization procedures. Regarding the fracture response, further work is required to understand the damage mechanisms and the appropriate size effects that must be considered, because the application of the material models and mechanical properties from the macro-scale experimental measurements are straightforward for the elastic and strength parameters, but not for the fracture properties.
3.4. Sensitivity Analysis Results
A sensitivity analysis of the relevant material and process parameters that influence the curing micro-residual stresses is presented in this section. After a preliminary study following previous literature results [
6], the material parameters that have a direct influence in the residual stresses are the CTE, the chemical shrinkage and the matrix elastic modulus. On the other hand, from the process point of view, since thermal gradients and transient effects are negligible at the RVE scale, the only relevant parameter is the cure temperature because it has a direct influence in the final curing level.
Figure 16 shows the results for the sensitivity analyses for each of the selected variables regarding the micro-residual stress level and the failure index. The failure index corresponds to the maximum value of the failure initiation criteria, given by Equation (18), over the matrix region. The results presented for the stress measure corresponds to the volumetric average over the entire matrix region of the RVE.
A clear tendency to increase the micro-residual stresses arises from increasing the CTE and the chemical shrinkage. For higher levels of micro-residual stresses, damage is achieved, with the failure index becoming higher than unity.
The elastic modulus also has a relevant influence on the micro-residual stresses. The tendency shows that micro-residual stresses increase with increasing elastic modulus, including scenarios of damage onset. It is worth to note that the strength properties are kept constant during the sensitivity analyses.
The cure temperature has a different influence on the micro-residual stresses because it changes the degree of cure evolution and the final curing level in the matrix and not the mechanical response directly. This difference in the curing level changes the final elastic modulus, Poisson’s ratio and overall mechanical behavior of the material. For lower temperatures, keeping the cure cycle time unchanged, the cure level is reduced to 20%, a very low value, but (as known) the final properties would not be suitable for a composite material.
Figure 17 shows the effective stress–strain relations, which present the expected tendency following the micro-residual stress analyses from
Figure 16. The identification of the curves corresponds to
Table 5; for example, CTE1 corresponds to the CTE value of case 1. The increase in micro-residual stresses generates a direct reduction of the material tensile strength.
The cure temperature reduction leads to a reduction in the elastic modulus followed by a loss of material strength (although resin strength parameters are kept fixed). The variation of the elastic modulus has a direct effect in the composite stiffness, as expected, and leads to a tensile strength reduction due to the higher micro-residual stress levels.
The sensitivity results show that the strength reduction can be mainly attributed to the micro-residual stress state that was characterized by the average von Mises stress and hydrostatic stress. Using all the data obtained above and plotting the tensile strength against the von Mises stress and the hydrostatic component, as shown in
Figure 18, two results are retrieved. First, the von Mises stress follows a linear relation with the average hydrostatic component for the curing micro-residual stresses state. Second, independently of the parameter that is being modified, the micro-residual stress effects hold, allowing us to conclude, within the scope of the current work, that the loss of material performance (tensile strength) is a direct consequence of the micro-residual stress levels.
Although it is recognized that additional work is required to appropriately understand the influence of process parameters on the mechanical response of the matrix resin at the micromechanical level, the difficulties associated to experimental work at this scale make the numerical insight a valuable tool to understand better the effective macro-mechanical response of composite materials, as demonstrated by the methodology proposed in this work.
4. Conclusions
The micro-residual stress analysis showed that stress concentrations are retrieved between the fibers in the narrower gaps, while the higher hydrostatic components are present in the resin-rich regions. This effect is clearly attributed to the constraining effect that the fiber arrangement imposes on the matrix, because the whole RVE is in traction-free condition.
During the curing process, the whole RVE experiences volume changes that follow the superposition of the CTE effects during heating, chemical shrinkage during curing, and the final thermal contraction during cooling down. This superposition gives a final volume reduction for the current example of 0.9%, approximately half of the chemical shrinkage (2%).
The micro-residual stresses have almost no influence in the effective stiffness of the composite system, but they have a more considerable effect in the strength properties, for example, reducing the transverse tensile strength by 11%. However, the experimental measured strength properties with macro-scale coupons are still slightly lower than the predicted properties considering the curing conditions.
The sensitivity analysis shows that increasing the CTE, chemical shrinkage and elastic modulus contributes to an increase of the micro-residual stresses. The cure temperature influences the final degree of cure of the material, degrading the elastic modulus.
A direct analysis of the transverse tensile strength as a function of the micro-residual stresses shows a linear relation between the von Mises stress and the hydrostatic stress. Therefore, the loss of performance can be attributed to the micro-residual stresses.