# Multiscale modeling of carbon fiber-reinforced polymer composites in low-temperature arctic conditions

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## Abstract

Exploration of new frontiers within the Arctic region introduces new challenges for the structural materials used in naval applications. This compels research on the influence of Arctic temperatures (from room temperature to \(-~70\,\,^{\circ }\hbox {C}\)) on the mechanical behavior of composites. In the current investigation, the effects of low temperatures on the axial stiffness of graphite/epoxy composites with unidirectional, cross-ply, and quasi-isotropic layups are studied using MAC/GMC, a micromechanical simulation tool developed by the NASA Glenn Research Center. Parametric studies were conducted to understand how various constituent material properties of a graphite/epoxy laminate influence the global, homogenized, axial stiffness of the composite subjected to arctic conditions. MAC/GMC provided accurate simulation results as compared with published experimental data. Results revealed that the increase in axial stiffness of carbon fibers is the main mechanism responsible for the overall increase in the global axial stiffness of the laminated composites at low temperature. The current research effort expands the understanding of how composites respond and behave in such extreme, low-temperature environments.

## Keywords

Carbon fiber-reinforced polymer (CFRP) Micromechanical modeling Modulus Low temperature Laminate layups## 1 Introduction

The Arctic region is experiencing sea ice shrinkage, and thus, new water ways for shipping and transportation routes are opening. Socially and politically, the geographic North Pole is not a territory of one single country. In fact, the Arctic consists of the Arctic Ocean and eight countries forming the Arctic states. Economically, the Arctic includes sizable natural resources, including 25% of the world’s undiscovered oil and natural gas (Arctic Resources 2018). The Arctic also holds 20% of the Earth’s fresh water supply. As such, exploration of the Arctic region is attractive to any state with intent to claim these natural resources, which are scarce in many parts of the world. However, the Arctic has demanding climatic challenges. Average winter temperatures can be as low as \(-~40\,\,^{\circ }\hbox {C}\), and the coldest recorded temperature is approximately \(-~68\,\,^{\circ }\hbox {C}\) (National Snow and Ice Data Center 2018). Coastal Arctic climates are moderated by oceanic influences, resulting in large-temperature variations. As a result, there is a demand to meet the material challenges to operate in such harsh climatic conditions.

Composite materials are being used frequently in various industries due to their impressive strength to weight and stiffness to weight ratios. These benefits successfully enabled advancement in aerospace applications such as the Boeing 787 Dreamliner which is manufactured with 50% of composite structures (The Boeing Company 2018). In terms of the effects of temperature, composites have primarily been used in high-temperature environments, such as in turbine blades and engine housings. Deeper exploration into the Arctic region, as described earlier, now offers a new and challenging area for composite usage. Thus, there is a resurging interest to investigate the effects of low temperature on composite materials.

Research has been performed in regards to compressive properties and compressive residual strength at low temperatures (Torabizadeh 2013; Sanchez-Saez et al. 2008). There are also several studies investigating the effects of low temperature on impact loading (Elamin et al. 2018; Salehi-Khojin et al. 2007; Suvarna et al. 2014; Im et al. 2001, as well as interlaminar delamination toughness (Coronado et al. 2014; Asp 1998; Coronado et al. 2012). Besides focusing on only low-temperature effects, researchers also studied the coupling effects of moisture and aqueous conditions on CFRP composites (Shen and Springer 1976; Rivera and Karbhari 2002. Adding to that, researchers have also investigated extremely low-temperature cryogenic conditions (Choi and Sankar 2006; Horiuchi and Ooi 1995; Gong et al. 2007; Schutz 1998; Schramm and Kasen 1977; Cease et al. 2006. However, these temperature ranges are significantly lower and inappropriate for application to the Arctic region. As with the papers on tensile properties, the above-referenced researchers do not explain the mechanisms that caused the change in composite properties at low temperature. It is clear that the fundamental understanding of composite stiffness, which forms the basis for other types of mechanical property studies, is far from being solved. There have been work done on the modeling of composites at low temperatures, particularly with the emphasis on how composites fail and damage at low temperatures (Shokrieh et al. 2012; Santiuste et al. 2011; Yang et al. 2015. However, the basis of mechanical properties was assumed as the bulk composite properties, without focusing on how individual constituent material property changes at low temperature. Other studies on the modeling of composites at low temperatures focus on the understanding of residual thermal stresses in composites (Shindo et al. 1993; Lord and Dutta 1988). Numerical tools are, indeed, useful to observe the distribution of stresses in the composite material.

Since there is now an increased demand to fully understand how and why composites behave at low-temperature arctic environments, this research aims to fill this gap by providing explanation for the change in tensile stiffness behavior of laminated composites at low temperature. The Micromechanics Analysis Code with Generalized Method of Cells (MAC/GMC) micromechanics software, developed by the NASA Glenn Research Center (GRC), is used herein to model repeating unit cells (RUCs) of composites considering the fiber and matrix phases explicitly (Bednarcyk and Arnold 2002a, b). The tensile behavior of graphite/epoxy composite RUC is simulated at low temperature and further validated vs. published experimental data. The simulation approach is often considered beneficial due to much reduced cost and time compared to experimental testing. Therefore, this study expands upon the limited available published experimental data shown earlier (Fig. 1) by predicting the global axial stiffness of various graphite-epoxy laminate layups in a low-temperature arctic environment. This research will provide insight to how graphite/epoxy composites behave in low-temperature arctic environments and how these results compare for different laminate layups and varying constituent material properties. These data are critical for the design of ship hulls for use in arctic exploration. Moreover, results from this work will delineate the role of various composite constituents in contributing to the composite global axial stiffness, allowing designers to tailor the composite material itself for optimal performance in an arctic environment.

## 2 Modeling approaches

In this work, the generalized method of cells (GMC) micromechanic theory, implemented within MAC/GMC, is used to calculate the effective properties of a composite RUC at various low temperatures, considering the fiber and matrix constituents explicitly. Below is a brief overview of the GMC theory. The reader is referred to reference Aboudi et al. (2013) for complete details on the formulation.

*h*,

*l*) by the following:

Carbon fiber and epoxy-matrix material properties used in the current study

Temp (\(^{\circ }\)C) | E, axial (Gpa) | E, trans (Gpa) | Nu, axial | Nu, trans | G (Gpa) | CTE Axial | CTE Trans |
---|---|---|---|---|---|---|---|

Fiber material properties | |||||||

23 | 219 | 12.4 | 0.29 | 0.29 | 20 | − 0.68 | 9.74 |

Epoxy material properties | |||||||

23 | 4.97 | 4.97 | 0.36 | 0.36 | 1.28 | 45 | 45 |

## 3 Results and discussion

Simulating the influences of temperature on the mechanical response of graphite/epoxy CFRP composites can be a challenging process. Accurate results are dependent on the validity of the constituent material properties of the composite. By understanding how various material constituent parameters affect the global composite axial stiffness, the most influential parameters that affect the global axial stiffness can be determined. These parameters are thus referred to as the dominant constituent properties. Once identified, the dominant constituent properties can then be calibrated to achieve the correct composite axial stiffness that correlates to experimental data. To determine the dominant constituent properties, a rigorous parametric study was performed with all constituent properties using MAC/GMC. By varying a single parameter from −10 to 10% of its original value, MAC/GMC can be used to predict the global, homogenized composite axial stiffness as a result of that parametric change. The procedure has been repeated for different constituent parameters and for various layup configurations (UD, CP, and QI). The results of the parametric study will identify and establish the *dominant constituent properties* influencing the composite axial stiffness of various layup configurations. Furthermore, simulation results will be validated with experimental data to back calculate the effect of low temperature on constituent material properties. To provide further insights, similar parametric approach will be employed to investigate how material constituent properties affect the global CTE of composite laminates. This section will end with a study on how extreme low-temperature cooling will create thermal residual stresses that might lead to matrix microcracking.

### 3.1 Unidirectional laminate simulation

A unidirectional (UD) graphite-epoxy composite is considered first. The RUC used in this study is ARCHID = 13, as shown in Fig. 3. A volume fraction of 0.65 is considered, as it is a common ratio used in the industry. To simulate the thermoelastic response of a composite laminate in MAC/GMC requires 14 material parameters: the axial and transverse moduli of the fiber and epoxy, the axial and transverse Poisson’s ratio of the fiber and epoxy, the axial and transverse coefficients of thermal expansion for the fiber and epoxy, and the shear modulus of the fiber and epoxy. Using material properties given in Table 1, these 14 parameters can be reduced, due to isotropy of the epoxy, to 10 parameters. This simplifies the user-defined properties needed to perform the parametric study to determine the dominant constituent properties of the composite.

Figure 5 shows the sensitivity to the global CTE of a UD to variations in the local thermoelastic properties of the constituents. Contrary to the stiffness, numerous local properties have a non-negligible influence on the global CTE. The most dominant property is the axial CTE of the fiber. A 10% increase in magnitude of this value resulted in a 67% increase in the magnitude of the global CTE from \(-~0.1\hbox {E}{-}6/^{\circ }\hbox {C}\) to \(-~0.167\hbox {E}{-}6/^{\circ }\hbox {C}\). Next, a 10% increase in the CTE of the matrix results in a 58% decrease in the magnitude of the CTE of the composite: \(-~0.043{E}{-}6/^{\circ }\hbox {C}\). The same variation in the stiffness of the epoxy yields a 56% decrease in the magnitude of the global CTE. Finally, 10% variation in the Poisson ratios of the fiber and epoxy leads to a 12% increase and 17% decrease in the global CTE of the composite, respectively. The sensitivity of a 10% change in the other properties is less than 1%.

### 3.2 Cross-ply laminate simulation

A cross-ply (CP) laminate was also simulated with a \([0/90]_{\mathrm{2s}}\) layup. Just like in the UD analysis described in the previous section, the same graphite-epoxy properties, which have effectively 10 different parameters, are used in the CP simulation. The same parametric study has been performed and the results are shown in Fig. 6a. The axial fiber modulus is clearly still the most influential constituent property, with regards to the effective axial stiffness of the laminate, in a CP layup. With a 10% variance in the fiber axial modulus, the overall composite axial stiffness is increased by 10% to around 82 GPa as opposed to the original room-temperature axial stiffness of around 74.5 GPa. To gain a better understanding on how the other parameters affect the overall axial stiffness of a CP laminate, the range of composite axial stiffness axis is again focused, as illustrated in Fig. 6b. The remaining parameters from most dominant to least dominant are as follows: the transverse modulus of the fiber, the modulus of the epoxy, and the Poisson’s ratios of epoxy and fiber. The shear moduli and coefficients of thermal expansion for the fiber and epoxy do not affect composite axial stiffness. The addition of the \(90^{\circ }\) fiber causes the variation in the Poisson’s ratio to affect the overall composite axial stiffness, as expected. A comparison between the UD and CP laminate studies shows some variation in the results. However, the same conclusion remains that the axial modulus of the fiber continues to be the most dominant constituent property with regards to the axial stiffness of graphite-epoxy composites.

Figure 7 shows the sensitivity of the global CTE of a CP laminate to changes in the local thermoelastic properties of the constituents. Similar to the UD, the CTE of the CP is sensitive to numerous local properties. The most dominant properties are the axial stiffness of the fiber and the CTE of the matrix. A 10% increase in the axial fiber stiffness results in a 10% decrease in the global CTE of the composite resulting in a change from \(2\hbox {E}{-}6/^{\circ }\hbox {C}\) to \(1.8\hbox {E}{-}6/^{\circ }\hbox {C}\). The 10% increase in the CTE of the matrix leads to an increase of the global CTE to \(2.2\hbox {E}{-}6/^{\circ }\hbox {C}\), or 10%. Next, a 10% increase in the Poisson’s ratio of the matrix produces a 9% increase in the global CTE. The same variation the stiffness of the matrix leads to a 7% increase in the global CTE of the laminate. A 10% increase in the transverse stiffness of the fiber results in a 5% increase in the CTE of the CP. Finally, a 10% increase in the axial and transverse CTE of the fiber leads to a 3% decrease and 2% increase in the global CTE, respectively. The influence of a 10% change in all other properties is less than 1% on the global CTE. It should be noted that, the effective CTE for CP, \([\pm 45]_{\mathrm{2s}}\), and QI laminates are identical, thus will not be discussed in subsequent sections.

### 3.3 Quasi-isotropic laminate simulation

The quasi-isotropic (QI) laminate considered in this study is a [45/–45/0/90]\(_{\mathrm{s}}\) layup. As previously described in the CP simulation, the ten material properties used as the epoxy and fiber material properties remain constant for all the various layup comparisons. The results of the QI parametric study are displayed in Fig. 8a. Supporting the trend observed in the previous simulations, the fiber axial modulus is, again, the most dominant constituent parameter. At 10% of its increased value, the overall composite axial stiffness was found to be 8.7% higher at around 56.5 GPa as compared to the regular room-temperature axial stiffness of around 52 GPa. This continues the expected trend that the lower the amount of fiber aligned with the loading direction, the lower the overall axial stiffness of the composite. More interestingly, the same 10% increase in fiber axial modulus results in the increase of composite axial stiffness by 10.4, 10, and 8.7% for the cases of UD, CP, and QI, respectively. This shows that the effect of fiber stiffness is becoming less dominant on the composite axial stiffness when the layup changes from UD to CP and further to QI. When the overall axial stiffness domain is enlarged in Fig. 8b, an interesting result is revealed when compared to the UD and CP layup simulations. The order of the remaining most dominant material properties to least is as follows: the shear modulus of the epoxy, the transverse modulus of the fiber, the epoxy modulus, the shear modulus of the fiber, and, finally, the Poisson’s ratio of the epoxy. The Poisson’s ratio does not affect overall composite axial stiffness. The QI laminate case is the first instance in which the shear modulus of the fiber and epoxy contributes to the variance in overall composite axial stiffness. This is due to the introduction of a non-orthogonal fiber orientations. The 45 and \(-~45^{\circ }\) fiber orientations result in shearing within the composite, which reduces the overall composite axial stiffness when compared to UD and CP laminates. Even with the effects of the shear moduli, the axial modulus of the fiber continues to be the most dominant constituent property in a QI laminate layup by far.

### 3.4 Validation with experimental results

Effect of temperature on carbon fiber axial modulus, by matching UD simulation results with UD experimental data (Kim et al. 2007)

Temperature \((^{\circ }\hbox {C})\) | Experimental composite axial stiffness (GPa) (Kim et al. 2007) | Simulation composite axial stiffness (GPa) | Percentage error between experimental and simulation axial stiffness (%) | Fiber, E axial used in simulation (GPa) |
---|---|---|---|---|

25 | 144 | 144.1 | + 0.07 | |

\(-\) 50 | 160 | 159.7 | \(-\) 0.19 | |

\(-\) 100 | 164 | 163.6 | \(-\) 0.24 | |

\(-\) 150 | 167 | 166.8 | \(-\) 0.12 | |

### 3.5 \([\pm 45]_{\mathrm{2s}}\) laminate prediction

### 3.6 Volume fraction influences

Since it utilizes micromechanics, MAC/GMC is capable of varying the volumetric fraction of the fiber. For all simulations so far, 0.65 volume fraction was assumed as it is a common ratio used in industry. The knowledge of the coefficients of thermal expansion of the fiber and epoxy gives insight to the effects of temperature on volume of those two materials. The parameters used are displayed in Table 1.

The fiber has about a zero coefficient of thermal expansion in the axial direction and a positive value in the transverse direction. This means that the fiber cross-sectional area will decrease with a decrease in temperature. The epoxy has a larger, positive coefficient of thermal expansion in both the axial and transverse directions. This concludes that as the temperature drops, the volume of the epoxy will decrease more than the fiber, thus increasing the overall volume fraction ratio of the composite.

### 3.7 Local residual stresses at extreme low temperatures

For all simulations in this section, the same RUC used in all previous simulations is used here, with a fiber volume fraction of 0.65. The thermoelastic properties of the matrix are temperature independent and given in Table 1. All of the carbon fiber properties, except the axial modulus, are temperature independent and presented in Table 1. The axial modulus is considered to be dependent on temperature and varies with temperature according to the calibrated results from Sect. 3.4, see Fig. 9.

Two separate macroscopic boundary conditions are considered. First, the global boundaries of the laminated composite are completely unconstrained, and all global boundary forces and moments are prescribed to be 0. These will be referred to as subjected to “stress-free cooldown.” For the second set of boundary conditions, global boundary deformations and rotations are fully constrained. Results from these two sets of boundary conditions should provide bounds on the actual in service boundary conditions of a composite panel integrated into the hull of an exploration vessel. Since the service loads are not known, no attempt is made to estimate these.

#### 3.7.1 Residual stresses in unidirectional laminate at extreme low temperatures

MAC/GMC is used to predict the residual stresses that arise during a cooldown from 23 to \(-~150\,\,^{\circ }\hbox {C}\) of a UD laminate under global stress-free boundary conditions. Figure 14 shows the residual stress in the UD RUC at \(-~150\,\,^{\circ }\hbox {C}\). Residual stresses are predicted considering a temperature-dependent axial modulus for the fiber (Fig. 14a), and all temperature-independent properties (Fig. 14b). In each subfigure within Fig. 14, the left plot is the local axial stress field, the middle plot is the local maximum principal stress field, and the right plot is the minimum principal stress field. The colorbars on the right of each contour plot show the value of the stress corresponding to the color in units of MPa.

Figure 15 shows the residual stress contours at \(-~150\,\,^{\circ }\hbox {C}\) for a UD laminate with constrained boundary conditions. Only temperature-dependent fiber properties are considered, since Fig. 14 indicates that there was a negligible difference in the residual stresses when comparing results considering temperature-dependent and temperature-independent axial fiber properties. With the constrained boundaries, the maximum axial residual stress in the fiber is − 28.3 MPa which is a reduction in the overall magnitude, as compared to the case with the stress-free boundaries. Conversely, the maximum principal stress in the matrix increases to 77.8 MPa when the boundaries are constrained. When comparing this to the typical range of matrix strengths, it is clear that the likelihood of matrix cracking in the epoxy is great in the UD. Finally, the minimum residual stress in the matrix is − 27.3 MPa, which is an increase in magnitude from the results considering stress-free cooldown.

#### 3.7.2 Residual stresses in cross-ply laminate at extreme low temperatures

Figure 16a, b displays the residual stress contours in the \(0^{\circ }\) ply of a CP laminate subjected to a cooldown from 23 to \(-~150^{\circ }\hbox {C}\) subjected to stress-free and constrained boundary conditions, respectively. The magnitude of the axial compressive stress increased substantially to a value of − 103.3 MPa in the stress-free case as compared to the UD. However, when the boundaries are constrained, the magnitude is reduced yielding a value of − 28.3 MPa. Although the magnitude of axial compressive residual stress in the CP with unconstrained boundaries is an order of magnitude greater than in the UD, it is still an order of magnitude lower than the typical compressive strength of a fiber. Thus, compressive fiber failure is not a concern. The maximum residual principal stress in the matrix is rather insensitive to the boundary conditions and was calculated to be 75.1 MPa, for after stress-free cooldown, and 77.8 MPa, when the boundaries are constrained during the temperature drop. Therefore, matrix cracking should be a major concern for a vessel that is to be used in arctic conditions utilizing CP or QI laminates (all plies in CP and QI laminates exhibit the same residual stress state when subjected to only thermal loads), which are more suitable for structural design than UD laminates. Residual stresses in the \(90^{^{\circ }}\) ply of the CP laminate are identical to those in the \(0^{^{\circ }}\) ply because of the orthotropic nature of the CP laminate and the absence of mechanical loading.

## 4 Conclusions

Through the use of MAC/GMC, a graphite/epoxy composite has been simulated at low-temperature representative of environmental conditions in the Arctic. This was done by first performing a parametric study on how the constituent material properties affect the overall composite axial stiffness. It was concluded that the axial fiber modulus in a unidirectional laminate contributes the most and thus can be solely manipulated to match already published experimental data. Cross-ply and quasi-isotropic laminates are also dominated by the axial fiber modulus and can, therefore, be simulated using the material properties found from the unidirectional results. Thus, over a temperature range from room temperature to \(-~150\,\,^{\circ }\hbox {C}\), a graphite/epoxy composite will increase in axial stiffness approximately by 15.7, 14.6, and 13.3% for a unidirectional, cross-ply, and quasi-isotropic laminate, respectively. A \([\pm 45]_{\mathrm{2s}}\) laminate cannot be simulated using the same assumptions as the laminate is shear dominated and no experimental data are available to validate this simulation. Results also reveal that a change in fiber volume fraction due to thermal expansion does not contribute enough to the overall composite global axial stiffness, and thus can be neglected during a low-temperature situation. Simulation results show that thermal residual stresses can be induced by extreme low-temperature cooldown, thereby possibly causing matrix microcracking. MAC/GMC has clearly shown that a graphite/epoxy composite can be simulated over a low-temperature range and has good agreement with experimental results. These results can, therefore, be used in the future design and simulation of composite layups at low-temperature Arctic conditions.

## Notes

### Acknowledgements

K.T. Tan acknowledges the research grant #N00014-16-1-3202 provided by the Office of Naval Research (ONR Program Manager: Dr. Yapa Rajapakse).

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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