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

  • Dominic R. Cross
  • K. T. Tan
  • Evan J. Pineda
  • Brett A. Bednarcyk
  • Steven M. Arnold
Original Paper


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.


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.

There exists limited published experimental results on the stiffness of carbon fiber-reinforced polymer (CFRP), specifically graphite fibers with epoxy-matrix system, at low-temperature conditions. Figure 1 summarizes the results from these experiments for three different laminate layups: unidirectional (UD), cross-ply (CP), and quasi-isotropic (QI). Kim et al. (2007) studied the effects of thermomechanical loading cycles of T700/epoxy UD laminate and concluded that axial tensile stiffness tends to increase as temperature decreases and the composite was hardly sensitive to the loading cycles after three cycles. Reed and Golda (1994) reviewed the mechanical and thermal properties of epoxy-matrix UD laminates reinforced with boron, alumina, aramid, S-glass, E-glass, and high strength, high modulus, and medium modulus carbon fibers from 4 to 295 K. Their data concluded that, at low temperature, the composite tensile strength and modulus depend primarily on the fiber strength and secondarily on the matrix resin. Majerski et al. (2012) studied tensile properties of UD AS7J carbon/epoxy M12 resin laminates, and found that the tensile stiffness of the laminate increases with decreasing temperature. There are much less published data for CP laminates. Sanchez-Saez et al. (2002) compared an AS4 carbon fiber embedded in 3501-6 epoxy-resin CP laminate to that of a QI laminate at low temperatures. Both laminates showed that the ultimate tensile strength decreased with temperature. However, the axial stiffness of CP laminates decreased with decreasing temperature, while that of QI laminates increases with decreasing temperature. Nettles and Biss (1996) performed mechanical testing on IM7/8551-7 carbon-fiber/epoxy-resin QI laminates, and concluded that low temperature negatively affects the impact behavior of the tape laminate due to the high in-plane thermal stresses. It is evident from Fig. 1 that there exists consistently increasing tendency in the axial stiffness for UD laminates when temperature decreases, but the trends for CP and QI are not clear. Moreover, the published papers do not provide explanation for the increase in laminate axial stiffness at low-temperature conditions.
Fig. 1

Collection of published experimental data on tensile axial stiffness of graphite/epoxy composites at low-temperature conditions

Fig. 2

Micromechanics links between structures and materials (NASA)

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

NASA GRC’s MAC/GMC software tool (Bednarcyk and Arnold 2002a, b) was utilized in this study for all micromechanic simulations. This general purpose code can be used to predict not only elastic and inelastic thermomechanical, but also smart material and piezo/magnetoelectric responses. The analysis code allows for the simulation of inelastic deformation, damage, failure, and fatigue life. A typical multiscale modeling hierarchy based on MAC/GMC is shown in Fig. 2. The composite RUC is composed of various constituent phases with different constitutive behaviors, and the RUC can be homogenized to predict the effective stress–strain behavior of a material point within a composite structure. A reference and further explanation for MAC/GMC can be found in MAC/GMC 4.0 User’s Manual—Keywords manual (Bednarcyk and Arnold 2002a) and Example Problem Manual (Bednarcyk and Arnold 2002b).
Fig. 3

RUC examples in MAC/GMC (Bednarcyk and Arnold 2002a, b)

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.

GMC is a micromechanical formulation for predicting the overall thermoinelastic behavior of the multiscale composites. There are four steps involved in this homogenization process. First, the RUC is identified and discretized. Figure 3 shows three different RUCs representing the same square-packed composite microstructure. A typical, doubly periodic RUC consists of \(N_\beta \times N_\gamma \) rectangular subcells (\(\beta =1,\ldots ,N_\beta ; \gamma =1,\ldots , N_\gamma )\) in the \(x_2 \) and \(x_3 \) directions, respectively, as shown in Fig. 3. The individual subcells have the dimensions of \(({h_\beta , l_\gamma })\) and are related to the unit cell dimensions (hl) by the following:
$$\begin{aligned} h=\mathop {\sum }\limits _{\beta =1}^{N_\beta } h_\beta ,\quad l=\mathop {\sum }\limits _{\gamma =1}^{N_\gamma } l_\gamma . \end{aligned}$$
As the second step, the relationships between the macroscopic average stresses and strains and the microscopic fields are established. A linearly displacement variable, \(u_i^{({\beta \gamma })} ({i=1, 2, 3})\), is considered inside each subcell. This displacement profile is consisted of pure displacement components in the center of each unit cell and the micro-variables representing the linear dependence of the displacement to the local coordinates.
The average strain in the RUC is defined as follows:
$$\begin{aligned} \bar{{\varvec{\varepsilon }}}_{{\varvec{ij}}} =\frac{1}{hl}\mathop {\sum }\limits _{\beta =1}^{N_\beta } \mathop {\sum }\limits _{\gamma =1}^{N_\gamma } h_\beta l_\gamma \bar{{\varvec{\varepsilon }}}_{{\varvec{ij}}}^{({{\varvec{\beta }} {\varvec{\gamma }} })}, \end{aligned}$$
which is based on the definition of strain tensor in each subcell \(\bar{{\varvec{\varepsilon }}}_{{\varvec{ij}}}^{( {{\varvec{\beta }} {\varvec{\gamma }} })} \) as:
$$\begin{aligned} \bar{{\varvec{\varepsilon }}} _{{\varvec{ij}}}^{({{\varvec{\beta }} {\varvec{\gamma }} })} =\frac{1}{2}({\partial _i u_j^{( {\beta \gamma })} +\partial _j u_i^{({\beta \gamma })} }). \end{aligned}$$
Considering a general constitutive equation for the thermoelastic material in each subcell \(({\beta , \gamma })\), the average stress in that subcell is calculated as follows:
$$\begin{aligned} \bar{{\varvec{\sigma }}}_{{\varvec{ij}}}^{({{\varvec{\beta }} {\varvec{\gamma }} })} ={\varvec{C}}_{{\varvec{ijkl}}}^{({{\varvec{\beta }} {\varvec{\gamma }} })} ({\bar{{\varvec{\varepsilon }}}_{{\varvec{kl}}}^{( {{\varvec{\beta }} {\varvec{\gamma }}})} -\bar{{\varvec{\varepsilon }}}_{{\varvec{kl}}}^{{\varvec{T}} ({{\varvec{\beta }} {\varvec{\gamma }}})}}), \end{aligned}$$
where the \({\varvec{C}}_{{\varvec{ijkl}}}^{({{\varvec{\beta }} {\varvec{\gamma }} })} \) is the elastic tensor, and \(\bar{{\varvec{\varepsilon }}}_{{\varvec{kl}}}^{{\varvec{T}} ({{\varvec{\beta }} {\varvec{\gamma }} })}\) are the average thermal strain tensor, respectively, in each subcell. Based on the volumetric summation of the average stresses in all subcells, the average stress for the RUC can be written as follows:
$$\begin{aligned} \bar{{\varvec{\sigma }}}_{{\varvec{ij}}} =\frac{1}{hl}\mathop {\sum }\limits _{\beta =1}^{N_\beta } \mathop {\sum }\limits _{\gamma =1}^{N_\gamma } h_\beta l_\gamma \bar{{\varvec{\sigma }}}_{ij}^{({{\varvec{\beta }}{\varvec{\gamma }}})}. \end{aligned}$$
In the third step, the continuity of displacement and traction is applied in an average, or integral, sense. It is considered that the RUCs are periodic and the interfaces of the subcells and the boundaries of the RUCs follow displacement and traction continuity. Applying these conditions results in elimination of the micro-variables in the displacement definition and produces a relationship between the macro- and micro-strain fields:
$$\begin{aligned} {\varvec{A}}_{\varvec{G}} {\varvec{\varepsilon }}_{\varvec{s}} ={\varvec{J}}\bar{{\varvec{\varepsilon }}} \end{aligned}$$
with \({\varvec{A}}_{\varvec{G}}\) and \({\varvec{J}}\) being matrices containing the geometrical details of the subcells and those related to the RUC, respectively. The \(\bar{\varvec{\varepsilon }}\) is the average macroscopic RUC strain, while the micro-strain vector \({\varvec{\varepsilon }}_{\varvec{s}}\) is defined as follows:
$$\begin{aligned} {\varvec{\varepsilon }}_{\varvec{s}} =({\bar{{\varvec{\varepsilon }}}^{( {\mathbf{11}})}, \ldots , \bar{{\varvec{\varepsilon }}}^{({{\varvec{N}}_{\varvec{\beta }}, {\varvec{N}}_{\varvec{\gamma }}})}}). \end{aligned}$$
Applying continuity of traction in terms of strains results in
$$\begin{aligned} {\varvec{A}}_{\varvec{M}} ({{\varvec{\varepsilon }}_{\varvec{s}} -{\varvec{\varepsilon }}_{\varvec{s}}^{\varvec{T}} })=0 \end{aligned}$$
in which \({\varvec{A}}_{\varvec{M}}\) includes the information regarding the material properties in each subcell, and \({\varvec{\varepsilon }}_{\varvec{s}}^{\varvec{T}} \) represents the thermal strains in each subcell and are defined in a similar way as Eq. (7).
Table 1

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







− 0.68


Epoxy material properties









The continuity conditions (Eqs. 6, 8) can be combined as follows:
$$\begin{aligned} \tilde{{\varvec{A}}}{\varvec{\varepsilon }} _{\varvec{s}} -\tilde{{\varvec{D}}}{\varvec{\varepsilon }} _{\varvec{s}}^{\varvec{T}} ={\varvec{K}}\bar{{\varvec{\varepsilon }}}, \end{aligned}$$
$$\begin{aligned} \tilde{{\varvec{A}}} =\left[ {{\begin{array}{c} {{\varvec{A}}_{\varvec{M}} } \\ {{\varvec{A}}_{\varvec{G}} } \\ \end{array} }} \right] ,\quad \tilde{{\varvec{D}}} =\left[ {{\begin{array}{c} {{\varvec{A}}_{\varvec{M}} } \\ 0 \\ \end{array} }} \right] ,\quad {\varvec{K}}=\left[ {{\begin{array}{c} 0 \\ {\varvec{J}} \\ \end{array} }} \right] . \end{aligned}$$
Solving Eq. (9) for each subcell
$$\begin{aligned} \bar{{\varvec{\varepsilon }}}^{\left( {{\varvec{\beta }} {\varvec{\gamma }} } \right) }={\varvec{A}}^{\left( {{\varvec{\beta }} {\varvec{\gamma }} } \right) }\bar{{\varvec{\varepsilon }}}+{\varvec{D}}^{\left( {{\varvec{\beta }} {\varvec{\gamma }} } \right) }{\varvec{\varepsilon }} _{\varvec{s}}^{\varvec{T}} \end{aligned}$$
provides the average strain in each subcell in terms of the macroscopic elastic and thermal strains, through mapping via strain concentration matrices \({\varvec{A}}^{\left( {{\varvec{\beta }} {\varvec{\gamma }} } \right) }\) and \({\varvec{D}}^{\left( {{\varvec{\beta }} {\varvec{\gamma }} } \right) }\).
In the fourth step, the overall macroscopic constitutive equations of the material based on the effective properties can be determined. The average stress in the RUC is given by the following:
$$\begin{aligned} \bar{{\varvec{\sigma }}} ={\varvec{B}}^{*}\left( {\bar{{\varvec{\varepsilon }}}-\bar{{\varvec{\varepsilon }}}^{{\varvec{T}}}} \right) , \end{aligned}$$
in which the \({\varvec{B}}^{*}\) represents the effective elastic tensor of the composite:
$$\begin{aligned} {\varvec{B}}^{*}=\frac{1}{hl}\mathop {\sum }\limits _{\beta =1}^{N_\beta } \mathop {\sum }\limits _{\gamma =1}^{N_\gamma } h_\beta l_\gamma {\varvec{C}}^{\left( {{\varvec{\beta }} {\varvec{\gamma }} } \right) }{\varvec{A}}^{\left( {{\varvec{\beta }} {\varvec{\gamma }} } \right) }. \end{aligned}$$
The average thermal strain tensor \(\bar{{\varvec{\varepsilon }}}^{{\varvec{T}}}\) is also obtained in a similar fashion.
To model composite laminates, MAC/GMC includes classical lamination theory (Hyer 2009) in which the response of each ply is modeled using GMC. For details, the reader is referred to references Bednarcyk and Arnold (2002a, b); Aboudi et al. (2013).
Fig. 4

a Parametric study of material properties affecting composite global axial stiffness of a unidirectional (UD) layup; b enlarged view

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.

The results of the UD laminate are presented in Fig. 4a. The axial fiber modulus is clearly the most dominating factor in determining the overall axial composite stiffness. With a 10% variance in the fiber modulus, the composite axial stiffness is increased by 10.4% to about 159 GPa, while the original room-temperature axial stiffness is around 144 GPa. This is due to the well-known fact that fibers in UD CFRP composite sustain most of the load in the axial direction. The carbon fiber performs best along the axial direction and the UD laminate configuration has enhanced stiffness in the axial direction. To gain a further understanding of how other parameters affect the axial stiffness of the composite, the range of composite axial stiffness axis was limited to 0.5 GPa, as shown in Fig. 4b. In this enlarged view, it is clear that the modulus of the epoxy is the second most dominant parameter. However, its influence is a small fraction of that of the axial fiber modulus. The remaining parameters have a negligible affect composite axial stiffness of a UD laminate. In particular note, the coefficient of thermal expansion (CTE) values will not affect the composite axial stiffness, and thus, they are not included in Fig. 4 and will not be discussed in subsequent sections. Therefore, it can be concluded that the axial fiber modulus of the composite is the most dominant constituent property that directly affects the axial stiffness of a UD graphite-epoxy composite.
Fig. 5

Parametric study of material properties affecting composite global axial CTE of a unidirectional (UD) layup

Fig. 6

a Parametric study of material properties affecting composite global axial stiffness of a cross-ply (CP) layup; b enlarged view

Fig. 7

Parametric study of material properties affecting composite global axial CTE of a cross-ply (CP) layup

Fig. 8

a Parametric study of material properties affecting composite global axial stiffness of a quasi-isotropic (QI) layup; b enlarged view

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

After performing the three parametric studies for UD, CP, and QI laminated composites, in the previous sections, it can be concluded that the axial modulus of the fiber dominates the overall elastic axial stiffness of the composite in all three cases. This simplifies the problem of characterizing the constituent properties for a graphite/epoxy composite with low-temperature effects. With only one dominant constituent property, MAC/GMC can be used to simulate how the composite changes with low temperature by only modifying the user-defined axial fiber modulus. To further predict how a graphite/epoxy composite behaves at low temperature, experimental data are needed to back calculate and validate simulation input data using MAC/GMC. There exist limited published reports that provide data on graphite/epoxy composite axial stiffness at low temperatures. The papers that compare axial stiffness to varying temperatures are most commonly referring to UD laminates. References on CP and QI laminates are relatively rare (Fig. 1). Kim et al. (2007) conducted tensile stiffness experiments with a unidirectional graphite-epoxy-matrix composite from room temperature to \(-~150\,\,^{\circ }\hbox {C}\). Based on their experimental results documented in the paper Kim et al. (2007), and the result of the parametric studies discussed earlier, the experiment can be simulated in MAC/GMC. A UD laminate model was created in MAC/GMC with material properties from the graphite fiber and epoxy at room temperature, as shown in Table 1. To match the experimental results from Kim et al. (2007), the axial fiber modulus was modified until the simulated overall composite axial stiffness agreed with the experimental overall composite axial stiffness at room temperature. This was repeated for all temperatures: 25, − 50, − 100, and \(-~150\,\,^{\circ }\hbox {C}\), as summarized in Table 2. The plot of fiber axial stiffness as a function of temperature is plotted in Fig. 9. This study reveals the relationship between fiber axial modulus with respect to low temperature. It is interesting to note that the rate of increase in fiber axial modulus is much higher between \(-~50\,\,^{\circ }\hbox {C}\) to room temperature \(25\,\,^{\circ }\hbox {C}\), but gradually slows down from −50 to \(-~150\,\,^{\circ }\hbox {C}\). This process of calibrating the simulation to the experiment verifies the material properties for the carbon fiber over the low-temperature range. The same material properties were then used in the CP and QI laminate models to predict how composite axial stiffness of different layups changes with a large-temperature drop. The normalized results against room-temperature data are shown in Fig. 10. As expected, UD laminate has the highest increase in axial stiffness, followed by CP and QI layups. The plot also displays the percentage change of overall composite axial stiffness over a \(175\,\,^{\circ }\hbox {C}\) temperature drop. The UD composite axial stiffness increased the most by 15.7%, followed by 14.6 and 13.3% for the CP and QI layups, respectively. By comparing these simulated results to already published experimental results, as plotted in Fig. 11, we can see a good agreement with experimental results. Besides similar trend lines for UD laminates, the prediction also agrees well with the CP and QI data from Sanchez-Saez et al. (2002), whereby the axial stiffness for CP is higher than QI for the same material system. Using the same material properties for the UD laminate, Fig. 11 demonstrates an increase in composite axial stiffness due to low temperature, which is the result of the increase in fiber axial modulus at low temperature (Fig. 9).
Table 2

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)




+ 0.07


\(-\) 50



\(-\) 0.19


\(-\) 100



\(-\) 0.24


\(-\) 150



\(-\) 0.12


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

Following the studies of UD, CP, and QI laminates, MAC/GMC was used to predict how a \([\pm 45]_{\mathrm{2s}}\) laminate is affected by the constituent fiber and epoxy material properties. The results of this simulation are only predictions as no published data are available relating to low temperature, graphite/epoxy composite, and a \([\pm \,45]_{\mathrm{2s}}\) layup. The results of the study are given in Fig. 12. This case (\(\pm \, 45^{\circ }\)) does not follow the trend portrayed by the previous three cases (UD, CP, and QI), with the ± 45 angle laminate axial stiffness now being dominated by the shear modulus of the epoxy. The remaining parameters from most dominate to least are as follows: shear modulus of the fiber, the axial modulus of the fiber, the transverse modulus of the fiber, the epoxy modulus, and, finally, the Poisson ratios of the epoxy and fiber. Clearly, if there were experimental stiffness data available for a \([\pm 45]_{\mathrm{2s}}\) laminate at low temperature, the MAC/GMC model could be correlated to that data by tuning the epoxy constituent shear modulus.
Fig. 9

Plot of fiber axial modulus against low-temperature range

Fig. 10

Thermal effects on normalized tensile axial stiffness of graphite/epoxy composites with different laminate layup configurations

Fig. 11

Thermal effect comparison with the published experimental data on tensile axial stiffness of graphite/epoxy composites with different laminate layup configurations

Fig. 12

Parametric study of material properties affecting composite global axial stiffness for a \(\pm \, 45^{\circ }\) angle laminate layup

Fig. 13

Effect of fiber volume ratio on composite axial stiffness with varying layup configurations

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.

To see the effects of the volume fraction on the axial stiffness of the composite, the axial stiffness of the composite was determined when volume fraction varies from 0.65 to 0.75 for the four laminate layups previously discussed. The simulation results are displayed in Fig. 13. Over the 10% increase of the volume fraction with constant room-temperature material properties, the overall composite axial stiffness increased by 14.8, 14.3, and 18.1% for UD, CP, and QI laminates. Note that the experimental results for overall composite axial stiffness increased around 15% over a temperature drop to \(-~150\,\,^{\circ }\hbox {C}\). For the simulation to represent the experimental results with constant material properties, the volume fraction used in MAC/GMC would have to increase 10%. A calculation of the actual volume change of the fiber and epoxy over a \(173\,\,^{\circ }\hbox {C}\) drop shows that the volume ratio merely changes from 0.65 at room temperature to 0.654 (0.6% increase) at \(-~150\,\,^{^{\circ }}\hbox {C}\). This concludes that the 10% volume fraction increase, resulting from thermal expansion, needed to match the experimental results is not a reasonable explanation for the increase in overall composite axial stiffness. However, the increase in material property variance of the fiber axial modulus is a much more realistic mechanism to increase tensile stiffness property. Therefore, the effect of volume fraction increase from thermal expansion can essentially be neglected in the simulation of a graphite/epoxy composite axial stiffness over a drastic temperature reduction.
Fig. 14

Residual stresses in UD laminate RUC subjected to cooldown from 23 to \(-~150^{{\circ }}\hbox {C}\) under global stress-free boundary conditions. Units of residual stress colorbar are MPa. a Temperature-dependent axial fiber modulus: axial stress (left), maximum principal stress (middle), and minimum principal stress (right). b Temperature-independent axial fiber modulus: axial stress (left), maximum principal stress (middle), and minimum principal stress (right)

3.7 Local residual stresses at extreme low temperatures

Since an application for this composite system is exploration of the extreme low-temperature environments, it is possible that the service temperature that the material is subjected to drops from 23 to \(-~150\,\,^{\circ }\hbox {C}\). Due to the mismatch in the coefficients of thermal expansion among the fiber and matrix, residual stresses will arise during the severe cooldown. In this section, the residual stresses that arise locally within the constituents of the UD and CP laminates as the temperature is lowered are predicted. Analysis of the residual stresses in the \([\pm \, 45]_{\mathrm{2s}}\) and QI laminates is omitted, because, under pure thermal loading, the local residual stress states in each ply of a \([\pm \, 45]_{\mathrm{2s}}\) and QI laminates are identical to those in the plies of a CP laminate. It should be noted that residual stresses cannot be predicted without a suitable micromechanics platform, because these stresses are internal to the material and are a facet of the microstructure and thermoelastic constituent properties.
Fig. 15

Residual stresses in UD laminate RUC subjected to cooldown from 23 to \(-~150^{{\circ }}\hbox {C}\) under global constrained boundary conditions. Units of residual stress colorbar are MPa: axial stress (left), maximum principal stress (middle), and minimum principal stress (right).

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.

It can be readily seen that the temperature dependence of the axial fiber modulus has a negligible effect on the residual stresses that accrue during cooldown. At \(-~150^{\circ }\hbox {C}\), the axial compressive stress in the fiber is − 34.0 MPa. Typical compressive strength of a carbon fiber ranges from 2000–2500 MPa, and therefore, there is no danger of the fibers failing during the cooldown Soden et al. (1998). The maximum principal stress in the matrix is an adequate measure to determine the propensity for microcracking (Pineda et al. 2013). The maximum principal stress in the matrix is 63.1 MPa at \(-~150\,\,^{\circ }\hbox {C}\). The typical tensile strength of epoxy matrix ranges from 69 to 80 MPa (Soden et al. 1998). Therefore, the residual stresses induced during entry into extreme low-temperature thermal conditions may lead to matrix cracking in the UD laminate. Finally, the typical compressive strength of epoxy is given as 120–250 MPa (Soden et al. 1998). The minimum principal stress in the matrix is − 20.6 MPa, so there is minimal threat of compressive failure. Moreover, the compressive stress is in a region of high tensile principal stresses. Thus, matrix tensile failure, i.e., microcracking, would be the dominant mode of damage induced by residual stresses.
Fig. 16

Residual stresses in \(0^{\mathrm{o}}\) ply of CP laminate subjected to cooldown from 23 to \(-~150\,\,^{{^{\circ }}}\hbox {C}\). Units of residual stress colorbar are MPa: axial stress (left), maximum principal stress (middle), and minimum principal. a Stress-free cooldown. b Constrained boundary

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.



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.


  1. Arctic Resources & Transportation Information System (ARCTIS) Oil and Gas (2018). Accessed 28 March 2018
  2. National Geographic, Arctic (2018). Accessed 28 March 2018
  3. National Snow and Ice Data Center (2018) All About Arctic Climatology and Meteorology. Accessed 28 March 2018
  4. Kim M, Kang S, Kim C, Kong C (2007) Tensile response of graphite/epoxy composites at low temperatures. Compos Struct 79:84–89CrossRefGoogle Scholar
  5. Reed RP, Golda M (1994) Cryogenic properties of unidirectional composites. Cryogenics 34:909–928CrossRefGoogle Scholar
  6. Majerski K, Surowska B, Bienias J (2012) Tensile properties of carbon fiber/epoxy laminates at low and room temperatures. Compos Theory Pract 12:182–185Google Scholar
  7. Sanchez-Saez S, Gomez del Rio T, Barbero E, Zaera R, Navarro C (2002) Static behavior of CFRPs at low temperatures. Compos Part B 33:383–390CrossRefGoogle Scholar
  8. Nettles AT, Biss EJ (1996) Low temperature mechanical testing of carbon-fiber/epoxy-resin composite materials. NASA Technical Paper 3663Google Scholar
  9. Torabizadeh M (2013) Tensile, compressive and shear properties of unidirectional glass/epoxy composites subjected to mechanical loading and low temperature services. Indian J Eng Mater Sci 20:299–309Google Scholar
  10. Sanchez-Saez S, Barbero E, Navarro C (2008) Compressive residual strength at low temperatures of composite laminates subjected to low-velocity impacts. Compos Struct 85:226–232CrossRefGoogle Scholar
  11. Elamin M, Li B, Tan KT (2018) Impact damage of composite sandwich structures in arctic condition. Compos Struct 192:422–433CrossRefGoogle Scholar
  12. Salehi-Khojin A, Mahinfalah M, Bashirzadeh R, Freeman B (2007) Temperature effects on Kevlar/hybrid and carbon fiber composite sandwiches under impact loading. Compos Struct 78:197–206CrossRefGoogle Scholar
  13. Suvarna R, Arumugam V, Bull D, Chambers A, Santulli C (2014) Effect of temperature on low velocity impact damage and post-impact flexural strength of CFRP assessed using ultrasonic C-scan and micro-focus computed tomography. Compos Part B 66:58–64CrossRefGoogle Scholar
  14. Im K, Cha C, Kim S, Yang I (2001) Effects of temperature on impact damages in CFRP composite laminates. Compos Part B 32:669–682CrossRefGoogle Scholar
  15. Coronado P, Arguelles A, Vina J, Vina I (2014) Influence of low temperatures on the phenomenon of delamination of mode I fracture in carbon-fibre/epoxy composites under fatigue loading. Compos Struct 112:188–193CrossRefGoogle Scholar
  16. Asp LE (1998) The effects of moisture and temperature on the interlaminar delamination toughness of a carbon/epoxy composite. Compos Sci Technol 58:967–977CrossRefGoogle Scholar
  17. Coronado P, Arguelles A, Vina J, Mollon V, Vina I (2012) Influence of temperature on a carbon-fibre epoxy composite subjected to static and fatigue loading under mode-I delamination. Int J Solids Struct 49:2934–2940CrossRefGoogle Scholar
  18. Shen CH, Springer GS (1976) Effects of moisture and temperature on the tensile strength of composite materials. Department of Mechanical Engineering. University of Michigan,Google Scholar
  19. Rivera J, Karbhari VM (2002) Cold-temperature and simultaneous aqueous environment related degradation of carbon/vinylester composites. Compos Part B 33:17–24CrossRefGoogle Scholar
  20. Choi S, Sankar B (2006) Micromechanical analysis of composite laminates at cryogenic temperatures. J Compos Mater 40:1077–1091CrossRefGoogle Scholar
  21. Horiuchi T, Ooi T (1995) Cryogenic properties of composite materials. Cryogenics 35:677–679CrossRefGoogle Scholar
  22. Gong M, Wang XF, Zhao JH (2007) Experimental study on mechanical behavior of laminates at low temperature. Cryogenics 47:1–7CrossRefGoogle Scholar
  23. Schutz JB (1998) Properties of composite materials for cryogenic applications. Cryogenics 38:3–12CrossRefGoogle Scholar
  24. Schramm R, Kasen M (1977) Cryogenic mechanical properties of boron-, graphite-, and glass-reinforced composites. Mater Sci Eng 30:197–204CrossRefGoogle Scholar
  25. Cease H, Derwent P, Diehl H, Fast J, Finley D (2006) Measurement of mechanical properties of three epoxy adhesives at cryogenic temperatures for CCD construction. Fermilab-TM-2366-AGoogle Scholar
  26. Shokrieh MM, Torabizadeh MA, Fereidoon A (2012) Progressive failure analysis of glass/epoxy composites at low temperatures. Strength Mater 44:314–324CrossRefGoogle Scholar
  27. Santiuste C, Barbero E, Miguelez MH (2011) Computational analysis of temperature effect in composite bolted joints for aeronautical applications. J Reinf Plast Compos 30:3–11CrossRefGoogle Scholar
  28. Yang P, Shams S, Slay A, Brokate B, Elhajjar R (2015) Evaluation of temperature effects on low velocity impact damage in composite sandwich panels with polymeric foam cores. Compos Struct 129:213–223CrossRefGoogle Scholar
  29. Shindo Y, Ueda S, Nishioka Y (1993) Mechanical behavior of woven composites at low temperatures. Fusion Eng Des 20:469–474CrossRefGoogle Scholar
  30. Lord H, Dutta P (1988) On the design of polymeric composite structures for cold regions applications. J Reinf Plast Compos 7:435–458CrossRefGoogle Scholar
  31. Bednarcyk B, Arnold S (2002) MAC/GMC 4.0 User’s Manual-Keywords Manual. NASA/TM-2002-212077/VolGoogle Scholar
  32. Bednarcyk B, Arnold S (2002) MAC/GMC 4.0 User’s Manual-Example Problem Manual. NASA/TM-2002-212077/VolGoogle Scholar
  33. Aboudi J, Arnold SM, Bednarcyk BA (2013) Micromechanics of composite materials: a generalized multiscale analysis approach. Elsevier Inc, New YorkGoogle Scholar
  34. Hyer MW (2009) Stress analysis of fiber-reinforced composite materials. DEStech Publications Inc, PennsylvaniaGoogle Scholar
  35. Soden PD, Hinton MJ, Kaddour AS (1998) Lamina properties, lay-up configurations and loading conditions for a range of fibre-reinforced composite laminates. Compos Sci Technol 58:1011–1022CrossRefGoogle Scholar
  36. Pineda EJ, Bednarcyk BA, Arnold SM, Waas AM (2013) Mesh objective progressive failure of a unidirectional fiber-reinforced composite using the method of cells. Int J Solids Struct 50:1203–1216CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe University of AkronAkronUSA
  2. 2.NASA Glenn Research CenterClevelandUSA

Personalised recommendations