Computational structural modeling and mechanical behavior of carbon nanotube reinforced aluminum matrix composites

Materials Science & Engineering A 614 (2014) 273–283

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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

Computational structural modeling and mechanical behavior of carbon nanotube reinforced aluminum matrix composites Yishi Su a, Zhiqiang Li a, Lin Jiang a, Xiaolu Gong b, Genlian Fan a, Di Zhang a,n a

State Key Laboratory of Metal Matrix Composites, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China Laboratory of Mechanical System and Simultaneous Engineering, University of Technology of Troyes, UMR CNRS STMR 6279, 12 Rue Marie Curie, 10010 Troyes, France

b

art ic l e i nf o

a b s t r a c t

Article history: Received 30 March 2014 Received in revised form 15 July 2014 Accepted 16 July 2014 Available online 23 July 2014

Due to their remarkable mechanical properties, carbon nanotube (CNT) reinforced aluminum (Al) matrix composites have attracted a wide range of research interests. This work attempts to experimentally and numerically investigate the relationship between the micro-structures and mechanical behavior of CNT/ Al composites. Three-dimensional (3D) computational structural modeling of CNT/Al composites is performed, in which the size, morphology, orientation, location and volume fraction of CNTs are reproduced to be similar to those of the actual micro-structures of CNT/Al composites. The strengthening of the mechanical properties of the constituent materials of CNT/Al composites and reasonable load and boundary conditions are studied based on the models of CNT/Al composites developed. The tensile mechanical behavior of CNT/Al composites is numerically evaluated and experimentally verified. Results show that the enhanced mechanical properties of CNT/Al composites can be attributed to three factors: CNT reinforcements, matrix grain refinement and layered architectures. Through the microscopic structural modeling methods presented herein, the effects of model size, interfacial behavior, volume fraction of CNTs and layered structures on the mechanical behaviors of CNT/Al composites can be reproduced to understand the strengthening and deformation mechanisms of CNT/Al composites. & 2014 Elsevier B.V. All rights reserved.

Keywords: CNT–metal composite Structural modeling Layered structure Interfacial behavior Mechanical behavior

1. Introduction Carbon nanotubes (CNTs) possess a unique combination of high stiffness, strength and tenacity [1,2], as well as superior thermal and electrical properties [3] for structural and functional material applications. It is generally believed that CNTs can be used as reinforcements to produce novel composites [4,5]. CNT reinforced polymer and metal matrix composites have been widely developed based on various types of matrix material [6,7]. However, in contrast to the intensive research on CNT/polymer composites, relatively few studies have been conducted on CNT/metal composites, particularly CNT/Al composites. Although the excellent performance of CNT/Al composites has been reported in several experimental investigations and CNT/Al composites are promising for the development of ultra-strong, lightweight materials [8–10], few numerical studies on this material have yet to be conducted, hindering a deep understanding of the phenomena relevant to the composite [11,12]. Herein, we establish a series of 3D microscopic structural models for CNT/Al composites, in which the size, morphology, n

Corresponding author. Tel.: þ 86 21 34202634; fax: þ86 21 34202749. E-mail address: [email protected] (D. Zhang).

http://dx.doi.org/10.1016/j.msea.2014.07.048 0921-5093/& 2014 Elsevier B.V. All rights reserved.

orientation, location and volume fraction of CNTs can be reproduced. For a single CNT, a detailed model is created based on statistical information gathered from numerous CNTs, such as the CNT diameter, length and orientation. The mechanical properties and interfacial behaviors of the constituent materials in CNT/Al composites are introduced to conduct uniaxial tensile tests, whereas fine meshes and proper load and boundary conditions are imposed to balance the computational accuracy and cost. With respect to the physical structures and processing factors of CNT/Al composites, the effects of model size, interfacial behavior, volume fraction of CNTs, and layered structure on the mechanical behavior of CNT/Al composites are simulated. The simulation results are in accordance with experimental data, indicating a remarkable reinforcing effect originating from the CNT structures. Overall, the 3D microscopic structural modeling method developed in this study is demonstrated to be an effective route to understand the strengthening and deformation mechanisms of CNT/Al composites.

2. Experimental procedures To prepare CNT/Al composites, commercial CNTs functionalized with carboxyl groups (–COOH) were purchased, and ball-milled Al

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Fig. 1. Preparation and experimental tests of CNT/Al composites: (a) ball-milled Al flakes, (b) dispersed CNTs on Al flakes, (c) 1.0 vol% CNT/Al BLC and (d) Al matrix grain size.

flakes ( 500 nm in average thickness) were produced from spherical powders (10 μm in diameter and 99.5% pure) [13], as shown in Fig. 1(a). The surfaces of Al flakes were then modified using polyvinyl alcohol (PVA) and uniformly mixed with the CNTs dispersed in a suspension, as shown in Fig. 1(b). The mixture of Al flakes and CNTs was then pre-heated in a flowing Ar atmosphere at 500 1C for 2 h to remove the PVA, compressed under a pressure of 500 MPa, sintered at 550 1C for 2 h, and cooled in a furnace to room temperature. Finally, the mixture was gradually heated to 440 1C and extruded to an extrusion ratio of 20:1 at a speed of 0.5 mm/min in a vacuum furnace. Therefore, CNT/Al composites denoted “biomimetic laminated composites” (BLCs) with a final compactness of over 99.5% were produced. Based on microscopic observations of the CNT/Al composites, the average thicknesses of the Al flakes were reduced to 400 nm; further details have been reported in a previous study [14]. After the preparation of the CNT/ Al composites, uniaxial tensile specimens of the composites with varying volume fractions of CNTs were prepared and machined. The tests were conducted at a strain rate of 5  10  4 s  1 at room temperature on a universal testing machine (AUTO-GRAPH AG-I, Shimadzu Co. Ltd., Japan). Fig. 1(c) presents the micro-structure of a 1.0 vol% CNT/Al BLC imaged on an optical microscope and a transmission electron microscope (TEM JEM-2100, JEOL, Japan), whereas Fig. 1(d) presents the grain size of the Al matrix in CNT/Al BLCs with CNT volume fractions of 0.5 vol%, 1.0 vol% and 2.0 vol%, respectively.

3. Numerical modeling 3.1. Computational structural modeling 3.1.1. Single CNT As shown in Fig. 2(a), CNTs are an allotrope of carbon that exhibit a cylindrical structure. For microscopic structural modeling, a single CNT can be treated as two coaxial cylinders with different radii [11], as shown in Fig. 2(b), where symbols Ro, Ri and l are the outer radius, the inner radius and the length of the CNT, respectively. Moreover, the flexural morphology of CNTs is considered as well by using a random factor. To evaluate the size distribution of CNTs, such as their diameter and length, a large number of CNTs were analyzed to achieve statistical results. Fig. 2(c) and (d) presents the length and diameter distributions of numerous CNTs, respectively. These results were measured from SEM and TEM images of randomly dispersed CNTs on Al flakes. Because the material between the outer and inner walls of single CNTs is simply graphite, a mathematical relation between the outer radius Ro and inner radius Ri can be established as follows: Ri ¼ ð1  ρCNT =ρG Þ1=2 Ro

ð1Þ

where pCNT and pG are the densities of CNTs and graphite, respectively. In this work, the density of CNTs and graphite was 1.60 g/cm3 and 2.25 g/cm3, respectively [15]. Based on the statistical results, the

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Fig. 2. Microscopic structural characteristics of CNT reinforcements: (a) TEM micrographs, (b) structural models, (c) length distribution (μm) and (d) diameter distribution (nm).

average length of the CNTs l is 700 nm, and the average outer radius of the CNTs Ro is set to 25 nm. Using Eq. (1), the average inner radius of the CNTs Ri is calculated to be 13.4 nm, which can be verified by experimental measurements [16]. Therefore, these structural characteristics of CNTs are applied to create the microscopic structural models of CNT/Al composites.

3.1.2. CNT/Al composites To generate 3D microscopic structural models of CNT/Al composites, we randomly distributed CNTs in an Al matrix. Fig. 3(a) presents a basic flowchart showing how to reproduce the actual composite structures of CNT/Al composites in which numerous CNTs are randomly dispersed. Fig. 3(b) shows the geometrical model of a single CNT, in which the average length LCNT, the outer radius Ro and inner radius Ri of a single CNT are based on the statistical results. Moreover, the random size factors of the length Lf and radius Rf of CNTs are applied to describe the geometrical variation in the morphology of CNTs. Fig. 3(c) presents the process by which numerous CNTs are randomly dispersed in CNT/Al composites, where the CNTs are created one by one, avoiding any overlap with the pre-generated CNTs. The increasing volume fraction of CNTs Vf is then compared with the desired volume fraction of CNTs Vf0 in real composites. Once the desired volume fraction of CNTs has been achieved, the construction of the 3D microscopic structural model of the CNT/Al composites is complete. Therefore, the size, morphology, orientation, location,

and volume fraction of CNTs are reproduced in line with the actual microstructures of CNT/Al composites. Fig. 3(d) and (e) presents the microscopic structural models (2 μm  2 μm  2 μm) of 1.0 vol% CNT/Al composites in which numerous CNTs are successfully dispersed in two dimensions (denoted as “BLC” in Fig. 3(d)) and in three dimensions (denoted as “BMC” in Fig. 3(e)). Not all of the CNTs overlap internally and may touch each other within the structural models of the CNT/Al composites. Moreover, to form the 3D microscopic structural models of the CNT/Al composites, a representative cubic model with side length L can cut the dispersed CNTs off. Therefore, the size distribution (such as the length and diameter) and the flexural morphology of the CNTs in the microscopic structural models of the CNT/Al composites are not similarly repeated each time because of the diversities of single CNTs and the cutting location. In addition, the desired volume fraction of CNTs Vf (calculated by the volume ratio of constituent materials in CNT/Al composites) can be precisely controlled by the structural modeling program and the volume fraction of CNTs based on the following equation: Vf ¼

nπðR2o  R2i Þl L3  nπR2i l

ð2Þ

where Vf, n, Ro, Ri and l are the volume fraction, the number, the outer radius, the inner radius and the length of the CNTs, respectively, and L denotes the side length of the CNT/Al composite models.

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Fig. 3. Computational structural modeling of CNT/Al composites: (a) flowchart of structural modeling, (b) single-CNT model, (c) dispersed-CNTs model, (d) 1.0 vol% CNT/Al BLC and (e) 1.0 vol% CNT/Al BMC.

3.1.3. Load and boundary conditions Fig. 4(a) and (b) presents the as-constructed structural models of 1.0 vol% CNT/Al composites with a side length L of 0.8 μm in which CNTs are distributed in two dimensions in a CNT/Al BLC and in three dimensions in a CNT/Al BMC, respectively. In Fig. 4(c), a cubic structural model of CNT/Al composites containing CNT reinforcements and an Al matrix is drawn in an XYZ Cartesian coordinate system. The fixed region X¼0 (Z¼0) shows 6 degrees of freedom (DOFs) fixed, and the moving region X¼L (Z¼L) constrained by the reference point “RF” can only translate with UX (UZ) DOF along the X (Z) loading direction. For all of the 3D structural models of the CNT/ Al BMC and BLC, the displacement loading about UX (UZ)¼ 0.12 L is imposed on the reference point “RF”. The general linear 3D solid tetrahedral element C3D4 (4-NODE) is used to mesh the 3D structural models of CNT/Al composites, as shown in Fig. 4(d). Four integration points are applied on a single element over the structural models. Average mesh sizes of 20 nm for the CNT reinforcements and 30 nm for the Al matrix are applied to generate fine grids. In this analysis, more than 200,000 elements are produced for each 3D structural model of the CNT/Al composites. In addition, the mechanical properties of the constituent materials are introduced into these microscopic structural models to simulate the mechanical behavior of the composites using the finite element code ABAQUS. The simulated results, e.g., the displacement UX (UZ) and the reaction force RFX (RFZ) experienced by the reference point “RF”, will be presented and discussed later in detail. 3.2. Mechanical properties of constituent materials 3.2.1. CNT reinforcement and Al matrix Due to their excellent mechanical properties, CNTs have been widely used as reinforcements in composite materials. The widely

accepted mechanical properties of CNTs were measured by Yu et al. [2], whereas catalytic CNTs with relatively low mechanical properties have been reported as well [1,6,8]. Considering the complexity and difficulty of determining the mechanical properties of CNTs, an elastic modulus E of 590 GPa, Poisson's ratio υ of 0.3 [10,11] and tensile strength σUTS of 30 GPa [16–18] are used here. For a pure Al matrix, an elastic modulus E of 70 GPa and Poisson's ratio υ of 0.33 are applied [19], as shown in Fig. 5(a). Regarding the yield stress of the metal matrix, the increase in yield stress with grain refining can be expressed by the empirical Hall– Petch relation [20,21], σ o ¼ σ i þ λd

 ð1=2Þ

ð3Þ

where σo and σi are the yield stress and the intrinsic stress resisting dislocation motion, whereas λ and d denote a constant and the grain size, respectively. For pure Al metal with grain sizes of 200 and 72 nm, the yield stresses are 238 and 283 MPa, respectively [9]. Based on Eq. (3), the values of σo and λ can be determined to be 170 MPa and 954.6 MPa nm1/2, respectively. Fig. 5(b) describes the yield stress of the Al matrix σο as a function of grain size d, which agrees well with the experimental results [22,23]. It should be noted that in CNT/Al composites, the grain size of the Al matrix should be smaller due to the presence of CNTs, and the average Al grain size in the CNT/Al composites (seen in Fig. 1(d)) was experimentally determined to be 2177 20.6 nm in this work. The uniaxial stress–strain relation of the Al matrix can be expressed as follows:
 Eεp n σ ¼ Aεn ¼ σ 0 1 þ ð4Þ σ0 where σ, A, n, σo and εp are the flow stress, the stress constant, the hardening exponent, the yield stress and the plastic strain, respectively. For the Al matrix, the stress constant A and the

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Fig. 4. Microscopic structural models of CNTs/Al composites: (a) 1.0 vol% CNT/Al BLC, (b) 1.0 vol% CNTs/Al BMC, (c) load and boundary conditions and (d) meshed structural model of 1.0 vol% CNT/Al BLC.

hardening exponent n were determined to be 320 MPa and 0.028, respectively. Combining Eqs. (3) and (4), the stress–strain relation of an Al matrix with an average grain size of 230 nm can be determined by multiplying by a yield stress factor, as shown in Fig. 5(c).

reinforcements and Al matrix [27], respectively. These parameters can be further expressed as follows:

3.2.2. Enhanced mechanical properties In CNT/Al composites, the enhanced mechanical properties contributed by the Al matrix can be mainly attributed to two factors: (i) Hall–Petch strengthening originating from grain size refinement; and (ii) dislocation strengthening due to the existence of CNT reinforcements [24]. In the previous section, the Hall–Petch strengthening was considered. Dislocation strengthening, which occurs as a result of the nucleation of dislocations or resistance to dislocation glide in a metal matrix, will also be considered in this section. Therefore, enhanced the stress–strain relation of a metal matrix after quenching hardening can be modified as follows:
 Eεp n σ ¼ Aεn þ Δσ dis ¼ σ 0 1 þ þΔσ dis ð5Þ σ0

Δσ EM GND ¼ 1:25Gb

Δσ OR ¼

Gb 1 d pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ln 2π 1  υ ð0:953= V f  1Þd b sffiffiffiffiffiffiffiffiffiffiffiffiffi 8V f ε0

Δσ CTE GND

bd

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Vf b ¼ 1:25Gb 12jΔCTEjjΔTj 1Vf d

ð7Þ

ð8Þ

ð9Þ

The increase in yield stress of the Al matrix Δσdis due to dislocation strengthening can be expressed via a quadratic relationship [25] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 CTE 2 ð6Þ Δσ dis ¼ ðΔσ OR Þ2 þðΔσ EM GND Þ þ ðΔσ GND Þ

where G, b and υ are the shear modulus, the Burgers vector and the Poisson ratio of the metal matrix, respectively, with ΔCTE representing the mismatch between the coefficients of thermal expansion, ε0 is the strain due to modulus mismatch, ΔT is the maximum change in temperature, and Vf, d and d are the volume fraction, the average diameter and the equivalent diameter of the CNT reinforcements, respectively. Eq. (5) clearly indicates that the increase in yield stress that occurs during quenching hardening depends on the size and volume fraction of CNTs. For the CNT/Al composites, G is 26.4 GPa, b is 0.286 nm, ΔT is 550 1C, ΔCTE is 22.6  10  6 K  1 [15], ε0 is 0.002, d is 50 nm, d is 137.95 nm and Vf is equal to 0.5%, 1.0% and 2.0%. Therefore, the presence of CNTs can increase the yield stresses of the Al matrix by 36.9, 63.8 and 91.1 MPa, as shown in Fig. 5(d).

where ΔσOR is the contribution of the Orowan strengthening CTE derived from the presence of CNTs [26] and Δσ EM GND and Δσ GND are the contributions of stress strengthening caused by the modulus mismatch and thermal expansion mismatch between the CNT

3.2.3. Behavior of CNT–matrix interface To evaluate the effect of the CNT–matrix interface on the mechanical properties of CNT/Al composites, we consider three

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Fig. 5. Mechanical properties of CNT reinforcements and Al matrix: (a) elastic properties, (b) Hall-Petch strengthening, (c) tensile stress–strain relations and (d) dislocation strengthening.

basic interfacial behaviors: (i) adhesion interface; (ii) friction interface; (iii) cohesive interface [28]. Such considerations are based on two factors: the microstructures observed in TEM images (Fig. 1(c)) and the CNT/Al phases that are sufficiently compatible to form a coherent embedding of CNTs in the matrix materials [29]. The adhesion interface is realized in the created structural models of CNT/Al composites, and the friction interface is constructed by considering a coefficient of friction of 0.1 [30]. For the cohesive interface, the cohesive zone model (CZM) can characterize the initiation and propagation of interfacial damage as an alternative criterion, in which the interfacial properties are specified by the interfacial strength t, separation distance δf and fracture energy Γ. For the case of linear softening behavior in the CZM, the damage factor D and fracture energy Γ can be defined as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2 tn ts D¼ þ ð10Þ t t Γ¼

1 f tδ 2

ð11Þ

where tn and ts are the interfacial stress components in the normal and tangential directions, respectively, and the CZM can be implemented using a FE code according to the surface-based contact behaviors. For the CNT–Al interface, the interfacial strength t is equal to 140 MPa [31], the separation distance δf is set to 0.5 nm [32] and the fracture energy Γ is determined to be 0.140 J/m2 [33]. Based on the three abovementioned CNT–Al

interfaces, the mechanical behaviors of CNT/Al composites can be numerically simulated.

4. Results and discussion 4.1. Stress–strain relations of CNT/Al composites with different composite structures and loading directions From the simulation results obtained for the CNT/Al composites, the stress–strain relations can be determined [34]. For each microscopic structural model of the CNT/Al composites with a certain volume fraction of CNTs, the tensile stress is calculated by dividing the reaction force RFX (RFZ) experienced by the reference point “RF” by the initial area L  L Stress ¼

Reaction force RF X ðRF Z Þ Initial area L  L

ð12Þ

In addition, the tensile strain of each microscopic structural model is determined by dividing the displacement UX (UZ) experienced by the reference point “RF” by the initial length L Strain ¼

Displacement U X ðU Z Þ Initial length L

ð13Þ

According to this definition, the numerical tensile stress–strain relations of 1.0 vol% CNT/Al composites with an adhesion interface can be determined. Fig. 6(a) presents the numerical tensile stress–strain

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Fig. 6. Tensile stress–strain relations and equivalent stresses in reinforcing CNTs at tensile strain of 0.1 for 1.0 vol% CNT/Al BLC composites with adhesion interface.

Fig. 7. Tensile stress–strain relations of 1.0 vol% CNT/Al BLC composites with adhesion interface under different loading directions.

relations of 1.0 vol% CNT/Al composites obtained by six different simulations. It should be noted that different structural models of CNT/Al composites were created in each simulation, i.e., it is not simply six repetitive computations of the same composite structural model, but six different simulations of the various structural models of CNT/Al composites. Although the composite structural models of CNT/ Al composites created are different, as shown in Fig. 6(b)–(g), the numerical tensile stress–strain relations obtained are generally similar to one another within the entire tensile strain region. For the CNT/Al composites with an adhesion interface, the equivalent stresses in only a portion of the CNT reinforcements at a tensile strain of 0.1 reach the tensile strength (30 GPa). This result indicates that microscopic

structural modeling can reproduce the composite structure and stable mechanical properties of CNT/Al composites, even for CNT reinforced MMCs. Fig. 7(a) and (b) presents the experimental and numerical tensile stress–strain relations of 1.0 vol% CNT/Al composites along different loading directions (X-, Y- and Z-directions). Generally, the experimental tensile stress–strain relation agrees well with the numerical results in the elastic region, and the tensile stresses appear to be somewhat lower than those of CNT/Al composites with an adhesion interface in the plastic region. Such results indicate that using the composite structural models of CNT/Al composites with an adhesion interface may over-estimate the

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Fig. 8. Tensile stress–strain relations and equivalent stresses in reinforcing CNTs at tensile strain of 0.1 for 1.0 vol% CNT/Al BLC composites with different model sizes.

Fig. 9. Tensile stress–strain relations and equivalent stresses in reinforcing CNTs at tensile strain of 0.1 for 1.0 vol% CNT/Al composites with different interfacial behaviors. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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composites' mechanical properties. Although the loading direction varies for the same composite structural model, there are small differences between the tensile stress–strain relations of the CNT/ Al composites due to the different orientations and locations of individual CNTs. Fig. 7(a) shows that the numerical true tensile stress–strain relations nearly coincide along the X- and Z-directions, whereas the numerical tensile stress–strain relations vary along the X- and Z-directions, as shown in Fig. 7(b). It should be noted that the numerical tensile stress–strain relations are always smallest along the Y-direction. Generally, although the composite structural models and loading directions of the CNT/Al composites are different, the composite structural models with an adhesion interface produce relatively uniform and overestimated effects on the mechanical properties of the CNT/Al composites.

4.2. Stress–strain relations of CNT/Al composites with different model sizes and interfacial behaviors Fig. 8(a) presents the numerical tensile stress–strain relations of 1.0 vol% CNT/Al composites with an adhesion interface along the X-, Y- and Z-directions, in which the model size L is 0.4, 0.8 and 1.2 μm (Fig. 8(b)–(d)). For these three model sizes, the numerical tensile stress–strain relations of the 1.0 vol% CNT/Al composites are generally in accord within the entire strain region, whereas small differences exist due to the variation in the morphology of the CNTs in the CNT/Al composites. This result indicates that a preferred model size of 0.8 μm can be used for the numerical simulations to effectively balance the computational accuracy and

281

cost. Moreover, there are small differences between the tensile stress–strain relations of the CNT/Al composites under different loading directions over the entire strain region, which again indicates the generally isotropic mechanical properties of the CNT/Al composites. Fig. 9(a) summarizes the experimental and numerical tensile stress–strain relations of 1.0 vol% CNT/Al composites with three different interfacial behaviors. The tensile stress–strain relation of the CNT/Al composite with an adhesion interface presents the largest stresses over the entire tensile strain region (0.0–0.12), whereas the tensile stress–strain relation of the CNT/Al composite with a friction interface yields the lowest stresses over the entire tensile strain region. Moreover, the tensile stress–strain relation of the CNT/Al composite with a cohesive interface presents results similar to those obtained for the CNT/Al composites with other interfacial behaviors. It should be noted that there is little difference between the tensile stress–strain relations of the 1.0 vol% CNT/Al composites with the three different interfacial behaviors, and the numerical results are generally in accord with the experimental results. Fig. 9(b), (c) and (d) presents the equivalent stresses at a tensile strain of 0.1 in the CNT reinforcements of CNT/Al composites with adhesion, cohesive and friction interfaces, respectively. Although the tensile strain of 0.1 is already large, the equivalent stresses (in blue color in Fig. 9(b)–(d)) in the majority of the CNTs are much lower than their tensile strength ( 30 GPa) for the CNT/Al composites with adhesion, cohesive and friction interfaces. Based on the results of these analyses, it can be observed that the CNT–matrix interfacial behavior plays a weak role in enhancing the mechanical properties of CNT/Al composites.

Fig. 10. Tensile stress–strain relations and equivalent stresses in reinforcing CNTs at tensile strain of 0.1 for 0.5 vol% and 2.0 vol% CNT/Al composites with different interfacial behaviors.

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Fig. 11. Tensile stress–strain relations and equivalent stresses in reinforcing CNTs at tensile strain of 0.1 for 1.0 vol% CNT/Al composites with different layered structures.

4.3. Stress–strain relations of CNT/Al composites with different volume fractions and layered structures Fig. 10 presents the experimental and numerical tensile mechanical properties of 0.5 vol% and 2.0 vol% CNT/Al composites with adhesion, cohesive and friction interfaces. For the CNT/Al composites with different volume fractions of CNTs, the experimental and numerical tensile stress–strain relations are in good agreement in the elastic strain region (ε o0.005). In the plastic strain region (ε 40.005), the CNT/Al composites with an adhesion interface exhibit slightly larger tensile stresses, whereas the CNT/ Al composites with a cohesive interface and friction interface present smaller tensile stresses. Once again, CNT/Al composites with varying volume fractions of CNTs and different interfacial behaviors produce very similar tensile stress–strain relations, which are generally consistent with the experimental results. Fig. 10(b) and (d) shows that the increasing equivalent stresses of the CNTs at a tensile strain of 0.1 reach their tensile strength (  30 GPa) with the increase in the volume fraction of CNTs (from 0.5 vol% to 2.0 vol%). In general, compared with the behavior of the CNT–matrix interface, the volume fraction of CNTs plays a more important role in enhancing the mechanical properties of CNT/Al composites. Fig. 11 presents the numerical tensile mechanical behaviors of 1.0 vol% CNT/Al BLCs (with the layer thickness denoted as “L  1” and the random layer thickness denoted as “L  2”) and CNT/Al BMCs (with completely random CNTs denoted as “R  1” and locally random CNTs denoted as “R 2”) with an adhesion interface. In Fig. 11(a), the tensile stress–strain relations of 1.0 vol% CNT/ Al composites in four cases (i.e., L  1, L  2, R  1 and R  2) are in

accord in the elastic strain region (ε o0.01). This finding indicates that 1.0 vol% CNT reinforcements can enhance the stiffness of the CNT/Al composites regardless of whether the CNTs are randomly dispersed in 2D or 3D. In contrast, in plastic strain region (ε 40.01), the 1.0 vol% CNT/Al BLCs with layered structures (both in L 1 and L  2 cases) present slightly larger tensile stresses than those exhibited by the 1.0 vol% CNT/Al BMCs with random structures (both in R  1 and R 2 cases). Fig. 11(b) and (c) presents the equivalent stresses of the CNT reinforcements in CNT/Al composites with different structures, in which the equivalent stresses of only a portion of the CNT reinforcements reach their tensile strength (  30 GPa). This finding suggests that the CNT/Al composites with layered structures can actually produce a greater enhancement in mechanical properties compared to that achieved by the CNT/Al composites with random structures.

5. Conclusions The computational structure modeling, experimental investigation and numerical simulation of tensile mechanical behaviors of CNT/Al composites were conducted in this work. In the numerical simulations performed, strengthening mechanical properties, layered structures and varying interfacial behaviors were considered. Several conclusions can be drawn. (i) Based on the microscopic structural characteristics of the constituent materials of CNT/Al composites, the size distributions of CNTs (such as length and diameter) are statistically summarized based on data gathered from numerous CNTs.

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Two flexural coaxial cylinders with different radii are used to create 3D microscopic structural models of single CNTs, in which random factors are introduced to realize the varying morphology and orientation of CNTs. The computational structural modeling program developed in this study can effectively establish 3D composite structural models of CNT/Al composites with randomly dispersed CNTs, whose size, morphology, orientation, lactation and volume fraction can be reproduced to reflect the actual microstructures of CNT/Al composites. (ii) After the preparation of CNT/Al composites with various volume fractions of CNTs, uniaxial experimental tensile tests of CNT/Al composites were carried out. By considering both the strengthening mechanical properties and the behavior of the CNT–matrix interface of the composite materials, numerical simulations of the tensile behavior of the CNT/Al composites with different volume fractions of CNTs were performed using the microscopic structural models developed. Based on the simulation results obtained for the CNT/Al composites, the numerical tensile stress–strain relations are well consistent with the experimental results. The findings indicate that compared with the behavior of the CNT–matrix interface, the volume fraction of CNTs plays a more important role in enhancing the mechanical behavior of CNT/Al composites. (iii) The developed structural modeling program can effectively establish microscopic structural models for the CNT reinforcements, Al matrix, CNT–matrix interface and layered structures of the composite materials considered in this study. The enhancement in the mechanical behavior of the CNT/Al composites increases with the volume fraction of CNTs in conjunction with the enhancement in the properties of the CNT–matrix interface and layered structure of the composites. The numerical simulation results obtained in this work can be used to establish the relationship between the composite structure and mechanical behavior of CNT/Al composites.

National High-Tech R&D “863” Program (No. 2012AA030311), National Natural Science Foundation of China (Nos. 51131004, 51071100 and 51271116) and Shanghai Science and Technology Committee (No. 14520710100).

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Acknowledgments [33]

The authors gratefully acknowledge the financial support provided by the National Basic Research “973” Program (No. 2012CB619600),

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