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Finite element method (FEM) is used to analyze the mechanical properties of carbon nanotubes (CNTs) reinforced polypropylene (PP) composites. Firstly, polypropylene is assumed as a viscoelastic material, while carbon nanotubes are assumed as linear elastic materials to study the effect of temperature on the mechanical properties of neat PP and CNT/PP nanocomposites. Secondly, to compare the viscoelastic properties of neat PP and CNT/PP nanocomposites, the relaxation time at a specific temperature is used to investigate the relaxation of the nanocomposites for fixed tensile displacements. Thirdly, the effect of CNT volume fraction on the viscoelastic properties of nanocomposites is studied at different temperatures. Finally, to better understand the stress distribution along the CNT axial direction, a single carbon nanotube is isolated in the matrix to compare the stress distribution with nonisolated CNTs.

Polypropylene (PP) is used in a wide variety of applications including packaging, textiles, and automotive components. Today, several interesting results have been reported for different PP structures: branching of linear Ziegler–Natta polypropylene [

As a low density material, PP is widely used in toys, package, decoration, etc. However, PP has lower elastic modulus and strength and poor impact resistance; to improve the mechanical properties of polymers, different types of particles with specific properties are often added into the matrix. Nano-clay with different content was blended with PP to increase the thermal and viscoelastic properties [

Although fillers can improve the viscoelastic properties of PP, nanoparticles are more interesting to modify the viscosity of PP due to their higher specific surface area [

Similar to mechanical improvement, CNT addition has a direct effect on the rheological properties of nanocomposites. The addition of CNTs can increase the polypropylene viscosity leading to a shift from a fluid-like to a solid-like behavior [

In our previous works [

A representative volume element (RVE) is used to evaluate the viscoelastic properties of PP-based composites. To study the mechanical behavior of this RVE via the finite element method, the effect of SWCNT content on the stress-time curves of SWCNT/PP nanocomposites under different temperatures is investigated. The numerical specimen is assumed to be a cylinder for which the length and end diameter are constant at 600 nm and 200 nm, respectively. SWCNT is randomly distributed in the cylinder (PP matrix) along the deformation direction. The SWCNT diameter is taken as 1.4 nm, and the length is 400 nm [

Numerical model for (a) mechanical model; (b) top view of the model; (c) side view of SWCNT/PP.

The commercial finite software MSC.MARC is used because of its wide range of element libraries [

MSC.MARC has two models to represent viscoelastic materials. The first is a Kelvin–Voigt model, while the second is a general hereditary integral approach. The Kelvin–Voigt model allows the rate of change of the inelastic strain to be a function of the total stress and the previous strain. The Kelvin–Voigt behavior (viscoelasticity) is modeled by assuming an additional creep strain. As represented in the Kelvin–Voigt model, the stress-strain relationships depend on the current stress and strain state. However, the complete stress history is necessary. In terms of the hereditary integrals or Duhamel integrals, the constitutive behavior is most readily expressed as these integrals are formed by considering the stress or strain build-up at successive times. Two equivalent integral forms exist: the stress relaxation form and the creep function form. Here, the stress relaxation form is used.

In the stress relaxation form, the deviatoric and volumetric behavior of an isotropic viscoelastic material are assumed to be fully decoupled and described by a time-dependent shear and bulk moduli. The bulk modulus is generally assumed to be time independent. However, this is an unnecessary restriction of the general theory. In this work, the bulk modulus is neglected, and the shear modulus can be expressed in a Prony series as given in [^{∞} represents the long-term shear modulus, and_{i} is a positive time constant (relaxation time). For short time (

Here, PP is assumed to be a viscoelastic material for the temperature range studied (20 to 80°C). When the temperature is 20°C, the relaxation modulus ^{n} is considered negligible, while the values for higher temperatures are taken from [

Finally, the SWCNT/PP nanocomposite is approximated as a two-phase material, and the SWCNT-PP interface is assumed to have perfect adhesion (no slip here, which will be studied in a future work). The mechanical properties of each component are listed in Table

Mechanical properties of the components.

Mechanical properties | SWCNT | Polypropylene |
---|---|---|

Young’s modulus (GPa) | 1030 | 1.88 |

Tensile strength (GPa) | 30 | 0.0348 |

Poisson ratio | 0.063 | 0.40 |

To characterize the viscoelastic behavior of the materials, the relaxation modulus is used to depict the time dependence of SWCNT/PP nanocomposites and neat PP. On the contrary, SWCNT content will play an important role on the mechanical properties of the nanocomposites. In these numerical simulations, the effect of SWCNT volume fraction on the viscoelasticity of the nanocomposites and neat PP is investigated.

In this work, the global mechanical properties of PP are selected [

Tensile reaction force-time curves of the numerical specimens for different temperature: (a) relaxation time

To clearly compare the effects of the temperature and relaxation time on the viscoelastic properties of composites, the specific loading time (0 s and 3 s) and CNT content (0.5%) are selected to analyze the results. Considering the cross-sectional area of the specimen, the reaction forces can be transferred to be the reaction stresses. The related data are plotted in Figure

Effects of temperature and relaxation time on viscoelastic properties of PP and CNT/PP with CNT volume fraction of 0.5%. (a) Temperature; (b) relaxation time.

Figure

Reaction force-time curves with different relaxation time: (a) neat polypropylene; (b) SWCNT/PP nanocomposites.

To better determine the effect of SWCNT on the mechanical properties of the neat polypropylene and nanocomposites at different temperature, the analyzed results of a single fiber (SWCNT) are isolated from the composite, so the stresses, strains, and other mechanical properties of the SWCNT can be separately determined. Figures

Results for one SWCNT in a nanocomposite for a relaxation time of

To better understand the stress distributions on the nodes, the stresses at different loading steps are presented in Figures

Figure

(a) Reaction force-time curves of PP and SWCNT/PP nanocomposites at node A for different temperature; (b) the displacement-arc length curves of the nodes along path AB at 80°C; (c) the position of nodes A and B; (d) the displacement-time curves of node C for SWCNT/PP nanocomposites when the load is fixed at 10 nN.

For a viscoelastic material, the deformation will increase if the load is constant. To analyze the viscoelastic properties for this case, a load of 10 nN is applied to the top of SWCNT/PP nanocomposites. The numerical results for the temperature selected are plotted in Figure

Assuming a relaxation time of 0.01 s, three SWCNT volume fractions (0.5, 1.0, and 1.5%) are selected to simulate the viscoelastic properties of SWCNT/PP nanocomposites and compare them with the viscoelasticity of the neat polypropylene. The numerical results for 20, 40, 60, and 80°C are plotted in Figures ^{o}C of PP is 122.8 MPa and relaxation occurs. When the time is 0.03 s, the reaction force of SWCNT/PP nanocomposites dropped to 24.251, 35.034, and 45.529 nN compared to 8.594 nN for the neat PP, and then the reaction forces do not change. When the temperature reaches 60°C and 80°C, the relaxation modulus of PP is 333.5 MPa and 462.1 MPa, respectively. In these two cases, the relaxation phenomena are more obvious, especially at 80°C; i.e., the relaxation modulus of PP is 462.1 MPa, and the reaction force of SWCNT/PP dropped from 49.230 nN to 28.675 nN with 1.5% of SWCNT. In general, with increasing temperature, the relaxation phenomena of the composites are more obvious with increasing temperature; i.e., higher variation levels over longer periods of time.

Reaction force-time curves of the nanocomposites with different SWCNT content for a 0.1 nm loading displacement at different temperatures: (a) 20; (b) 40; (c) 60; (d) 80°C.

To compare the effects of the relaxation time on the viscoelastic properties of SWCNT/PP nanocomposites with different SWCNT content, a relaxation time of 1 s is assumed. The results for 20, 40, 60, and 80°C are plotted in Figures ^{o}C, the reaction force of SWCNT/PP in Figure

Reaction force-time curves of the nanocomposites with different SWCNT content for a 0.1 nm loading displacement: (a) 20; (b) 40; (c) 60; (d) 80°C.

In this work, the relaxation time and relaxation modulus were used to depict the viscoelastic properties of polypropylene (PP) as a matrix. Then, the effects of temperature (20–80°C) and single-wall carbon nanotube (SWCNT) content (0.5–1.5 vol.%) were investigated using numerical simulations (finite element method) to determine the viscoelastic properties of PP and SWCNT/PP nanocomposites for uniaxial tension loading. From the results obtained, the main conclusions are as follows:

Keeping the loading displacement constant, the reaction forces decreased with time (relaxation). As expected, the changes were faster with increasing temperature and decreasing relaxation time. When the loading time is fixed 0 s and 3 s, the SWCNT volume fraction is fixed 0.5%, it is clear that the stress capacity of PP or CNT/PP decreases with increasing temperature, especially the temperature is taken as 80°C. Similarly, if the relaxation time is greater enough, the fluid-like characteristic of composite is more close to the solid-like characteristic.

Whatever the temperature and relaxation time, SWCNT can very efficiently improve the mechanical properties of the nanocomposites (rigidity) when the loading displacement is fixed, even at low content. However, increasing SWCNT volume fraction led to higher stiffness and strength of the SWCNT/PP nanocomposites. On the contrary, when the loading force was fixed, SWCNT efficiently restricted the deformation of the nanocomposites. As SWCNT content increased, the creeping behavior of the nanocomposites became less important.

By isolating a SWCNT in the matrix, it was possible to show that SWCNT can undertake more stresses than the neat PP in SWCNT/PP nanocomposites. At the same time, comparing the homogeneous material, SWCNT is randomly distributed in the matrix, so the stresses distribution in the matrix changes for different SWCNT locations. The nodes rigidly connected to SWCNT undertook more stresses than those of the neat PP regions.

All data, models, and code generated or used during the study appear in the published article.

The authors declare that they have no conflicts of interest.

This study was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant number: LY18E080028) and funded by the National Natural Science Foundation of China (Grant number: 51568009) and the Wenzhou Science and Technology Project, China (Grant number: S20190001).