A new theoretical creep model of an epoxy-graphene composite based on experimental investigation: effect of graphene content

Abstract

In this paper, the creep behavior of an epoxy-based adhesive reinforced with different weight fractions of graphene up to 0.5 wt% was studied. Creep tests were performed in three stress levels, using the ultimate strength of the neat epoxy as a reference. Results indicated that the presence of graphene up to 0.5 wt% reduces the creep strain and strain rate of the epoxy. However, the dominant behavior in the creep of epoxy–graphene composites is the creep pattern of the neat epoxy. These experimental observations led to development of theoretical creep models to an appropriate creep model for graphene-reinforced composites by introducing a new function of the graphene weight ratio. A scanning electron microscopy analysis indicated that the strong bond between the graphene surface and epoxy matrix limits the mobility of the molecular chains of the neat epoxy and therefore reduces the creep strain.

Keywords

Creep epoxy–nano-graphene composite theoretical model

Introduction

The use of nano-technology to reinforce polymers has greatly increased in the last decade, driven by the wider availability of high-performance nano-materials. Many research works have demonstrated that the addition of a small amount of nano-reinforcing materials to a neat polymer increases its mechanical properties dramatically.¹ A comprehensive review of the mechanical, thermal, and electrical properties of the polymers reinforced with micro or nano-fillers can be found in Corsby and Lee,¹ Giv et al.,² Bhattacharya,³ Subawi,⁴ Barbosa et al.,⁵ Ajeesh et al.,⁶ and Borghei et al.⁷ Various models have also been introduced to predict changes in the mechanical properties of the improved polymers, including strength, Young’s modulus and fracture toughness,⁸ and creep in polymeric composites.⁹ The accuracy of these models was later confirmed by the experimental results of Kinloch and Taylor.¹⁰

Polymers are known to suffer from creep even at low stress levels and normal temperatures (near room temperature) due to the nature of their constituent molecular chains and their mobility,¹¹ which can lead to unwanted deformation and even failure in long-term loading. Using polymer-based composites instead of neat polymers is an attractive solution to overcome this problem. Regarding to the relatively low strength and high creep of thermoplastic polymers compared to thermoset polymers, special focus was devoted first to improving the performance of thermoplastics. Among these studies, the use of carbon nano-tubes (CNTs) to enhance the creep behavior of the polypropylene^12–14 and polyurethane¹⁵ and the use of nano-clay to strengthen the low-density polyethylene¹⁶ can be mentioned. Afterwards, a multi-stage hierarchical micromechanical model was introduced by Hassanzadeh-Aghdam et al.¹⁷ to predict the creep behavior of polypropylene/CNT composites. They also used a three-dimensional unit cell-based micromechanical model based on nonlinear viscoelastic analysis to predict the creep and recovery behavior of a nano-reinforced PMMA/CNTs¹⁸ and polyimide/SiO₂¹⁹ nano-composites. With the widespread use of graphene in the polymer nano-composites, the effect of this material on the creep behavior of polymers was also studied. Tang et al.²⁰ studied the creep and recovery behavior of polystyrene by considering three different reinforcement nano-particles, including reduced graphene oxide, CNT, and carbon black. The results of their experiments showed that reduced graphene oxide has a much better effect on improving the creep behavior of polymer compared to other two nano-materials.

Compared to thermoplastic polymers, thermoset polymers have superior properties, such as strength, hardness, and thermal stability,²¹ and due to their highly cross-linked structure, they have higher creep resistance than thermoplastics polymers.²² The use of nano-material generally improves the mechanical properties of thermoset polymers;^1,2 however, the available reports on the creep behavior of these polymers with different reinforcing materials are rather spare. Zandiatashbar et al.²³ studied the effect of adding different amounts of nano-carbon material, including single-wall and multi-wall CNTs graphene, on the creep properties of epoxy, and concluded that the two types of CNTs had no effect on the creep resistance of epoxy, and that graphene shows the best effect in reducing creep strain. This effect is negligible at room temperature and low stresses but intensifies with the increase of temperature or stress level. Ghajar et al.²⁴ investigated the effect of various parameters such as the amount of CNTs on the viscoelastic behavior of epoxy-based nano-composites using the dynamic mechanical thermal analysis (DMTA) and creep test. According to the obtained results, increasing the content of CNT increases the creep resistance of the nano-composite. Khoramishad and Ashofteh²⁵ used a single lap joint to study the shear creep behavior of a neat, 0.1 wt%, and 0.3 wt% CNT-reinforced epoxy adhesive. They concluded that 0.1 wt% of CNT-reinforced epoxy adhesive shows the highest improvement of the shear creep resistance of the adhesive joint. Based on the reports of Starkova et al.,²⁶ the addition of MWCNTs up to 1 wt% has a negligible effect on the elastic, viscoelastic, and viscoplastic response of the epoxy system. Zhang et al.²⁷ also studied the effect of three different carbon nano-structures, including single-wall CNTs, multi-wall CNTs, and Fluorine (C60) on the behavior of epoxy-based nano-composites. Their experimental results indicated that SWCNT and C60 have a better effect on epoxy creep resistance, but MWCNT did not show much improvement in epoxy nano-composites. In another work, an empirical model was developed to describe creep behavior of PMR-15 polyamide resin following a transient and steady state creep model of metals and alloys at elevated temperatures.²⁸

So far, significant studies have been performed on the mechanical behavior of polymer-based composites. Despite the dispersion of the existing studies on the creeping behavior of thermosetting polymers reinforced with various nano-materials, the studies on graphene-reinforced nano-composites are in the same alignment. In this paper, the effect of multi layered graphene nano-platelets on the creep behavior of a thermoset epoxy adhesive has been studied with a new approach. The different values of nano-graphene and different stress levels have been considered, and then, based on the creep model of the neat epoxy, a new update was introduced for the creeping model of epoxy–graphene nano-composite. In addition, scanning electron microscopy (SEM) analysis of the fracture planes was performed to indicate if the presence of graphene nano-platelets increases the strength of epoxy–graphene composite by blocking the crack growth in reinforced specimens.

Materials and methods

Manufacturing procedure

An epoxy adhesive, with the trade name of Araldite 2011, Salt Lake City, Utah, EUA, was used for the research presented in this paper. This is a two-component epoxy adhesive (AW106/HV953U). To act as the reinforcement, a multi-layer (5–10 layers) graphene powder with 3 nm average thickness and 7.2 µm lateral size and trade name of av-PLAT-7 was used (Avanzare Innovación Tecnológica S.L., Logroño, Spain). The steps of preparing the reinforced epoxy adhesive using graphene nano-platelets are schematically shown in Figure 1. Since the viscosity of the hardener is less than the viscosity of the epoxy portion, nano-graphene was first added to the hardener part and mixed with a DAC 150.1FVZK (Hauschild Engineering, Germany) centrifugal mixer, for 5 min at 2800 r/min until a homogenous mixture of hardener and nano-graphene was achieved. Then, in the next step, epoxy resin was added to the mixture of the hardener-nano-graphene and mixed with a centrifuge mixer for 4 min at 2800 r/min.

After this step, the reinforced mixture of epoxy–graphene composite was considered to be ready for pouring inside a rectangular mold, equipped with a silicone rubber frame. In order to obtain high-quality samples without any air bubbles and voids, the mold was first pre-heated up to 60℃ in an oven. The reinforced adhesive mixture was then poured and spread uniformly throughout the mold. In the next step, the mold was placed for 45 min inside the hot plate press machine at a pressure of 2 MPa and at a temperature of 100℃ to ensure complete cure. This procedure lowered the viscosity of the adhesive and, due to the high curing pressure, expanded the silicone rubber frame that surrounds the adhesive, creating a hydrostatic pressure state the gathers micro-bobbles in the local areas and the removes some of the trapped air, leading to a practically void free and high-quality adhesive plate in the most areas. The molding process is based on the French NF T 76-142 ²⁹ standard, as described in da Silva et al.,³⁰ as shown in the Figure 2. The bulk samples with the uniform thickness of 2 mm were obtained from the cured plates of the epoxy adhesive after cutting through the shown dumbbell shape geometry in Figure 3, by a computer numerical control machine. It is worth to mention that a full cure of a two part epoxy takes 7–14 days. However, adequate strength for further assembly or packaging can be reached within minutes or hours.³¹ Here, the specimens were tested at least after seven days of preparation.

Test procedure

To assess the mechanical behavior of the neat and reinforced epoxy adhesive, bulk samples were subjected to tensile testing following the ASTM D638 standard.³² An Instron extensometer with an accuracy of ± 0.5% and 25 mm of gauge length was used to measure the strain during the test. The ambient temperature was 27℃, and the loading speed was 5 mm/min for all the specimens under simple tensile test.³² After the tensile tests were completed, a SEM analysis was performed to study the morphology of the graphene platelets and the fracture surfaces of neat and graphene reinforced epoxy.

The creep tests of the neat and reinforced bulk samples were performed following the ASTM D2990 standard.³³ A new experimental set up was created for this purpose, shown in Figure 4. In this set up, the force was applied using direct weight, and the displacements were measured through the use of an extensometer attached to the sample. In order to apply the intended load, the upper crosshead, which was attached to the specimen, was moved very slowly upwards and, after about 5 s, was completely separated from the lower surface, resulting in the entire hanging weight being applied to the specimen. At this exact moment, the creep test was started, with the load being applied by the hanging weight and the attached extensometer recording the total strain of the bulk sample every 2 s. According to equation (1), the creep strain curves are obtained by subtracting the total strain from the static strain. It should be mentioned that, in the experimental creep tests of this paper, the load remains constant during the time but not the stress. However, the stress amount here is considered equal to the load per initial cross section area.

ɛ_{cr} = ɛ_{total} - ɛ_{static}

(1)

Results

Mechanical properties

Representative stress–strain curves obtained from simple tensile tests are shown in Figure 5. According to this figure, it is clear that the addition of graphene nano-platelets to the epoxy adhesive up to 0.5 wt% leads to an increase in the strength as well as a reduction in the elongation at break or ultimate strain of the epoxy adhesive. The mechanical properties of the epoxy adhesive reinforced with nano-material are provided in Table 1. The data shown in this table demonstrate that the addition of nano-graphene up to 0.5 wt% to the epoxy adhesive increases the Young’s modulus and the strength of epoxy adhesive by 75% and 27%, respectively, reducing the ultimate strain by 49%.

Table 1.

Mechanical properties of neat and reinforced epoxy.

	Young’s modulus (MPa)	Strength (MPa)	Ultimate strain (%)
Neat epoxy	1022 ± 100.1	25.5 ± 1.3	45.1 ± 4.0
0.1 wt% Reinforced	1126 ± 92.3	26.6 ± 1.1	41.8 ± 4.0
0.25 wt% Reinforced	1399 ± 110.5	29.2 ± 0.9	28.9 ± 1.5
0.5 wt% Reinforced	1786 ± 132.3	32.3 ± 0.8	21.9 ± 1.7

Creep behavior of neat and modified epoxy

The creep test was carried out at three constant loading levels of 70, 150, and 220 N, which are equal to engineering stress levels of 4.4, 9.4, and 13.8 MPa, at the temperature of 27℃. Since the mean ultimate load of the neat bulk specimens are equal to 408 N, these three stress levels are almost equal to 17%, 37%, and 54% the ultimate strength of the neat epoxy adhesive. Figure 6 shows the creep strain diagrams of the neat and reinforced epoxy samples in the three constant load levels. As expected, with an increase in the load level, the creep strain rate increases in the adhesive samples. It can also be seen from these diagrams that graphene has a positive effect on the creep behavior of epoxy, noticeably reducing the strain and strain rate of this polymeric material.

Microscopic analysis

SEM images were taken of the graphene powder, as well as of the fracture surfaces of bulk samples of neat and nano-reinforced epoxy adhesive. These images are useful for a morphology analysis and an assessment of the distribution of the nano-platelets. Figure 7 shows the SEM images of different parts of used graphene nano-material. From these images, it can be understood that the nano-graphene has irregularly shaped surfaces. This nano-material consists of one or several layers of graphene, which is shown in Figure 7(a) and in some cases has a porous multi-layer structure (Figure 7(b)).

Figure 8 shows the fracture planes of bulk specimens made of (a) neat adhesive, (b) 0.25 wt%, and (c) 0.5 wt% graphene-reinforced adhesive. By analyzing the SEM images for all three samples, it was found that the quality of the samples was satisfactory, as no voids were observed at the fracture surface of the samples. By evaluating the fracture morphology of the neat and reinforced adhesive samples, it can be concluded that all of these samples have flat and smooth layers with sharp edges that indicate the brittle fracture in the cross section of neat and reinforced specimens. On the other hand, by increasing the graphene content in the epoxy adhesive, the amount of these fracture layers in the fracture surface increases.

In agreement with the SEM images in Yao et al.,³⁴ the neat sample exhibited a rather smooth fracture surface with less fracture layers; with increasing graphene content, up to 0.5 wt% of the epoxy adhesive, as shown in the Figure 8(c), the fracture layers increase and lead to a rough fracture surface with widespread irregular patches. It can be inferred that the existence of graphene nano-platelets with random and discontinuous dispersion in the structure of nano-composites can prevent the failures and blocks the growth of capillary cracks in the material, increasing the strength of the reinforced specimens.³⁵ On the other hand, the larger number of fracture layers, which occurs with the addition of nano-graphene from 0 to 0.5 wt%, increases the fracture surface area and improves the joint failure load. One possible interpretation could be that the presence of graphene content in the epoxy matric stops the crack growth, as it reaches to the surface of the hard graphene layer, which makes the fracture plane to be more layered and have irregular shape in comparison to the neat epoxy adhesive’s one. Therefore, the final strength and the elastic modulus increase in the nano-composite structure.

In general, nano-materials have a very high-specific surface area, which creates very large contact surfaces in nano-composites.³⁶ However, it should be noted that the addition of nano-material does not always improve the properties of formed nano-composites. If agglomerates are formed, these can act as defects resulting in a reduction in composite’s mechanical performance rather than any enhancement. This is a major challenge when preparing adhesive formulations.³⁷ In general, it can be said that the improvement in the mechanical properties of polymers is dependent on two factors: the reaction between the nano-material and polymer matrix, as well as the dispersion of nano-material in the polymer matrix.³⁸ Nevertheless, obtaining a nano-composite with stable dispersion of graphene and no agglomerations has always been a major challenge.³⁹

By analyzing the SEM images of the fracture surfaces of nano-reinforced epoxy adhesive, multi layers of graphene with a thickness up to 40 µm can be observed; in Figure 9, samples of these plates for the reinforced nano-composite with 0.25 wt% (Figure 9(a) and (b)) and 0.5 wt% (Figure9(c)) of the graphene can be observed. Also, it can be seen that, in general, the fracture plane of nano-composite with 0.5 wt% graphene is rougher than the 0.25 wt% epoxy–graphene nano-composite.

The fracture surfaces near the porous multi-layer graphene on the reinforced epoxy, as shown in Figure 9, can be a good benchmark for examining the quality of the bond between the surfaces of the graphene and the epoxy matrix, since they represent the boundary between graphene and polymer. The red arrows indicated in these figures show the fracture edges on the contact surface of the multi-layer graphene with the epoxy matrix. This type of fracture mechanism near the epoxy and graphene contact surfaces can indicate a strong bond between graphene nano-platelets and epoxy. It also can be inferred that the existence of this strong bond between the surfaces of the nano-graphene and the epoxy polymer limits the movement of the epoxy molecular chains, which reduces the elongation at break as well as time-dependent deformations in the structure of the epoxy–graphene composite. Despite the positive effects of nano-graphene in the structure of epoxy–polymer, it should be noted that agglomerated dispersion of the plates act as a defect in the material.³⁷ Also, since the molecular bonds between the single graphene platelets are of a weak van der Waals type,³ it can be expected that, using the low layer graphene and exfoliated dispersion, further improvement in mechanical properties of epoxy can also be achieved.

Theoretical approach

Creep model of neat epoxy adhesive

Creeping strain at constant temperature is a function of stress σ (which equals to constant load divided to the initial cross section area $(σ = P / A)$ ) and time t variables, and according to the time and stress superposition creep models, can be considered as the product of stress and time functions, as in equation (2). So far, several models have been introduced for these functions; the detailed description of these models can be found in Boyle and Spence.⁴⁰ In this paper, a nonlinear power law’s creep model of Norton–Bailey (equation (3)) is used to predict the creep behavior of the epoxy. This is a flexible model which can fit the creep strain, as the primary creep behavior is dominant.⁴¹ In equation (3), C, m, and n are the material constants which are related to the creep behavior of the material. m and n are dimensionless, and C has the dimension of [ ${Pa}^{- n} . s^{- m}$ ]. To obtain the material constants here, a nonlinear regression method with the root mean square error of 0.002275 and R² equal to 0.9996 was utilized. By a time differentiation of equation (3), the creep strain rate, another format of the creep formulation obtains (equation (4)). Figures 10 and 11 indicate the fitting of Norton–Bailey model on the experimental creep strain as well as the creep strain rate (based on equations (3) and (4)), respectively. As it can be seen in the figures, the both forms of the Norton–Bailey model provides a good pattern for fitting the empirical data for the creep of neat epoxy adhesive.

ɛ_{cr} = f (σ, t) = f_{1} (σ) . f_{2} (t)

(2)

\begin{matrix} ɛ_{cr} = C σ^{n} t^{m} C = 1.746 e - 6, n = 2.87, m = 0.5723 \end{matrix}

(3)

\begin{matrix} {\overset{\cdot}{ɛ}}_{cr} = (C . m) σ^{n} t^{m - 1} = C' σ^{n'} t^{m'} \\ C' = 0.9992 e - 6, n' = 2.87, m' = - 0.4277 \end{matrix}

(4)

Creep model of epoxy–graphene nano-composites

To model the creep behavior of the epoxy-based composite at a constant temperature, it can be assumed that the creep strain is a function of the stress $(σ)$ , time $(t)$ , and weight percentage of the nano-graphene in the structure of the composite $(Gr)$ , and this can be written in the form of equation (5). Using this equation, creep strain of the neat epoxy is obtained from equation (6). Now, if the creep strain of epoxy–graphene nano-composite is considered to be dependent on the neat epoxy creep strain, equation (7) can be introduced for the creep of epoxy–graphene composite. Function of $g (σ, t, Gr)$ in this equation is the normalized function of creep strain of the epoxy-based nano-composite based on the neat epoxy creep strain, which shows the effect of adding different contents of the nano-graphene to the epoxy. In general, this function can be a function of time and stress, as well as graphene weight percentage. According to equation (7), in order to obtain the normal function of $g (σ, t, Gr)$ , the creep strain of nano-composites with different weight percentage should be divided into the creep strain obtained from neat epoxy.

ɛ_{cr} = F (σ, t, Gr)

(5)

ɛ_{cr_neat} = F (σ, t, 0) = f (σ, t)

(6)

ɛ_{cr} = F (σ, t, Gr) = f (σ, t) . g (σ, t, Gr)

(7)

Another important point that can be inferred from the creeping pattern of neat epoxy and its graphene composites is that the neat epoxy creeping behavior is the dominant behavior in the creeping pattern of epoxy–graphene nano-composites. Figure 12 shows the $g (σ, t, Gr) | σ_{1}, σ_{2}, σ_{3}$ functions (normalized creep strain functions) at three stress levels for different weight fractions of the nano-graphene. According to these diagrams, it can be concluded that the creep strain of the epoxy–graphene composite initially is a function of time, but eventually reaches a steady pattern. In other words, after a short time (about 1000 s), the normalized creep strain will have a fixed pattern over time and can be assumed to be constant. In fact, it can be said that after this initial period, the normalized creep function gis an independent function of time. Therefore, the normalized creep strain function will be a function of stress and graphene weight ratio, or $g = g (σ, Gr)$ .

To investigate the stress effect on the normalized creep function, the mean values of this function after t = 1000 s can be compared for all three stress levels. These data are provided in Table 2. As it can be seen on the last column of the table, the constant amount of the normalized strain function in every load level has a variance of about less than ± 4% of its mean value. Therefore, if its dependency on the stress variable is assumed to be negligible, the normalized strain function can be considered as the function of graphene content in the epoxy-based composite, and therefore, $g = g (Gr)$ . The mean values of the $g (σ, Gr) | σ_{1}, σ_{2}, σ_{3}$ function for three stress levels are shown as solid dots in Figure 13.

Table 2.

Mean value of normalized creep strain function after t = 1000 s, for different graphene content.

Normalized creep strains	$σ_{1}$	$σ_{2}$	$σ_{3}$	Mean and variance values
Neat epoxy	1	1	1	1
0.1 wt% Gr reinforced	0.920	0.840	0.891	0.884 ± 4.1%
0.25 wt% Gr reinforced	0.807	0.765	0.804	0.792 ± 3.4%
0.5 wt% Gr reinforced	0.670	0.680	0.640	0.663 ± 3.5%

Gr: graphene.

To introduce an appropriate equation for predicting the behavior of

g (x = Gr)

function, four functions were used as trend lines: (a) exponential function

e^{ax}

, (b) fraction function

a / (a + x)

, (c) quadratic function

{ax}^{2} + bx + 1

, and (d) summation of hyperbolic sine and hyperbolic cosine,

\sinh (ax) + \cosh (bx)

. The constants of a and b for the desired functions, as well as the sum of the squared errors, are obtained for each of these functions. As it can be deduced from Figure 12 and Table 3, all four functions can describe the change in the pattern of epoxy behavior by adding graphene, but the two quadratic and

\sinh (ax) + \cosh (bx)

functions have better definition than the exponential and fraction functions.

Table 3.

Four trend lines for normalized creep function with constants, RMSE, and R².

Trend lines for g function	Constants	RMSE	R²
$e^{ax}$	$a = - 0.8492$	0.02831	0.9396
$a / (a + x)$	$a = 0.9823$	0.02313	0.9597
${ax}^{2} + bx + 1$	$a = 0.7424, b = - 1.03$	0.02413	0.961
$sinh (ax) + cosh (bx)$	$a = - 1.042, b = 1.289$	0.02401	0.9614

RMSE: root mean square error.

Conclusions

In this work, the effect of low graphene contents on the creep behavior of epoxy adhesive was studied. For this purpose, bulk samples were manufactured using neat and epoxy reinforced with 0.1, 0.25, and 0.5 wt% of graphene. The results of quasi-static tensile test showed that with the addition of nano-graphene until 0.5 wt%, the strength and the Young’s modulus of epoxy adhesive continuously increased by 27% and 75%, respectively; however, the ultimate strain was found to reduce by 49%. The SEM images indicated that the fracture surface of the nano-composite is very rough and irregular. Also, due to the presence of numerous failure layers around the agglomerated graphene in SEMs, it can be considered that there is a strong bond between the graphene and epoxy interface. This strong bond limits the movement of molecular chains and, as a result, reduces the ultimate strain.

The results of the creep test showed that the addition of nano-graphene up to 0.5 wt% reduces the creep strain in the material continuously. Epoxy-based composites containing 0.5 wt% graphene, at different stress levels, had a creep of 0.64 to 0.68 percent of the neat epoxy specimens, which can be due to the high strength of graphene nano-platelets as well as the presence of a strong bond between graphene and epoxy surfaces. To model the neat epoxy creep behavior, the Norton–Bailey model was used as an instance, and the corresponding creep constants were obtained with a mean square error of 0.002275.

The creep results of neat and graphene-reinforced epoxy indicated that the presence of graphene nano-platelets until 0.5 wt% reduces the creep strain of the neat epoxy adhesive constantly. It was observed that the normalized creep strain at various stress levels becomes uniform after a short period of creep time (about 1000 s). Also, the dependence of the function to the stress was investigated, and it observed that the function has negligible stress dependency. Four curves, including exponential function, fraction function, quadratic as well as summation of hyperbolic sine and hyperbolic cosine function, were used as a tend lines to fit the data. In conclusion, the newly presented creep model for graphene-reinforced epoxy, using the described ‘g’ functions, can be extended to other well-fitting creep models as well, since the normalized creep data, neglecting the type of creep model, has been used to extract it.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Bashir Behjat

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