On interfacial stress transmission of carbon nanotubes/alumina composites

The interface of carbon nanotube (CNTs)/alumina ceramic composites has a very important effect on their mechanical properties. In this study, an appropriate theoretical cell model was established to study the interfacial stress transmission in CNT/alumina composites. The stress transfer equation is derived as follows: The tensile stress of the CNTs and interfacial shear stress were simulated under an axial tension load, and the relationship between the stress transmission and the effective length of the CNTs was analyzed. The theoretical results were compared with the finite element method (FEM) results, and the results were found to be in good agreement. This study can provide a theoretical basis for adjusting the appropriate length of CNTs and interfacial interactions under different tension loads.


Introduction
CNTs have been recognized as an ideal reinforcement for alumina composites because of their excellent mechanical and electrical properties [1][2][3]; however, the excellent properties of CNTs cannot guarantee that the reinforced alumina composites will achieve the expected performance. The main reason is that interfacial characteristics have important effects on the mechanical properties of CNTs and their reinforced composites [4,5]. In CNT-reinforced composites, CNTs are easily pinned in the grain interiors. The fracture mode of the materials changes from intergranular fracture to transgranular fracture owing to the effect of CNTs' bridging and pinning; thus, more energy can be absorbed [6][7][8]. In addition, the fracture toughness of the materials was strengthened by the broken and pulled-out CNTs. For the reinforcement of brittle materials, the combined effect of the interfaces plays a major role in the enhancement of CNTs embedded in the alumina ceramics. The weak bound, which is the major factor for CNT-reinforced brittle materials, can enhance the fracture toughness of composites to a certain extent [9][10][11][12].
Experimental methods, numerical simulations, and continuum mechanics were used to study the CNTreinforced composites. Chowdhury et al used a molecular dynamics model to simulate the mechanical behavior of CNTs pulled out from a polymer [13]. Considering the effect of the matrix density, interfacial bonds, and geometric defects of CNTs, the relationship between the interfacial shear stress, density of the matrix, and chemical bonds between interfaces were investigated. Tserpes et al studied the tensile properties and hardness of CNT-reinforced composites using a multi-scale model [14], which includes the three-dimensional elastic beam model of the C-C bond, the structure model of CNTs, the CNTs equivalent beam model, and the representative volume element model. Uddin studied the mechanical properties of CNT/nickel metal composites by simulating multi-walled CNTs with three different diameters [15]. Shojaie et al investigated the trends in the elastic modulus and Poisson's ratio along the axis and transverse directions under different matrix moduli and interfacial binding energies, respectively [16]. Using a model in which CNTs were pulled out from the polymer, Li et al investigated the effects of CNTs length, diameter, and wall layers on the pulling-out process [17]. The interfacial shear stress is found only at the ends of the nanotubes and is independent of the length of the nanotubes; however, it is proportional to the diameter. Later, Liu et al studied the interfacial mechanical properties of CNT/alumina composites using the same method and obtained the same trends of interfacial shear stress [18].
In a theoretical investigation, Ko et al used a modified Cox model to investigate the stress transfer efficiency of single-walled CNT/epoxy composites and determined the effects of the length and diameter of single-walled CNTs on the stress transfer efficiency of composites [19]. Hague et al established a theoretical model to analyze the stress transfer of CNT-reinforced composites [20], which included the stress transfer equation, stress distribution, and effective length of the CNTs. In addition, the distributions of the axial stress and interfacial shear stress were analyzed using the shear lag theory. Jiang et al developed a cohesion model based on van der Waals forces between the interfaces [21]. The continuous medium model was used to simulate the matrix, and the relationships between the interfacial stress and interfacial displacement were determined. Lv et al studied the relationship between stress and displacement between multi-walled CNTs and the polymer matrix, as well as the mechanical effects of the number of layers, radius, and volume fraction of multi-walled CNTs on composites [22]. Chen et al considered the lattice effect and elastic properties of CNTs, and the load transfer between CNTs was theoretically analyzed [23]. The results showed that the utilization ratio of the strength of the nanotubes did not continue to increase when the length of the CNTs was five times greater than the characteristic length. Yan et al conducted microscopic interfacial stress analysis of CNT-reinforced alumina ceramic composites based on a modified COX model [24].
This paper is being a continuation of Yan's study [24], the difference between current study and previous study by Yan and Wei is that this paper discusses the distributions of the normal stress and shear stress near the interfaces, as well as the effective length of the CNTs in the alumina ceramic composites, finally, the proposed results were compared with the FEM results.
2. Molecular structure and interfacial mechanical model of CNTs/alumina composites 2.1. Structure of CNTs CNTs can be seen as a single layer or as multi-graphite sheets curling around a certain direction (figures 1(a) and (b)). Vector C h , which is the circumference of the CNTs, can be obtained when any two nodes on the graphite sheet are connected. The structure of the CNTs was uniquely determined by vector C h . If a 1 and a 2 are defined as unit vectors of the lattice structure of graphite, then C h = n a 1 + m a 2 , where n and m are the chirality parameters of CNTs. When n = m, CNTs are armchair type, and when m = 0, CNTs are zigzag type. In addition, nanotubes are helical because the grid of carbon atoms is a spiral. The planar density of the CNTs atoms is where l C is the equilibrium distance between two carbon atoms.

Lattice structure of an alumina matrix
α-alumina belongs to the trigonal system of R-3D space group and its single-cell model is sharp hexahedral, as shown in figure 2(a). The most stable crystal form of alumina is α-alumina in the nature, such as natural corundum, ruby, sapphire, etc α-alumina has a compact structure and low activity. Atoms O 2are arranged in the light of a hexagonal close-packed, which is story repeat type. Atoms Al 3+ are filled in the 2/3 octahedral according to Pauling ionic rule in which the distance between the ions is the farthest. The structural arrangement of α-alumina from the above description can be got. The bulk density of oxygen atoms and aluminum atoms of alumina crystals can be derived by unit cell of a-alumina and the lattice constants are a = 0.4759 nm, b = 0.4759

Interfacial mechanical model of CNTs / alumina ceramic composites
The crystal structure of alumina was arranged according to the type of element, as shown in figure 3(a). The alumina crystals were decomposed into aluminum (figure 3(b)) and oxygen (figure 3(c)) clusters. CNTs were uniformly dispersed in the alumina ceramic composites. A representative volume element (RVE) was selected, and interfacial mechanical analysis was performed ( figure 4(a)). To facilitate the study of interfacial interaction, the research models were extracted separately from the non-boundary interfaces and boundary interfaces. The power stroke of cohesion among atoms ranges from a few atoms to tens of atoms, and the non-boundary interfaces can be modeled as the interaction between infinite graphite sheets and atom clusters ( figure 4(b)). Similarly, atom clusters and graphite sheets with a free edge along the y axis can be used to model the interaction at the boundary interfaces (figure 4(c)).
In CNT-reinforced alumina ceramic composites, CNTs, as nanometer short fibers, are embedded in an alumina ceramic matrix. Assuming that CNTs are uniformly distributed in the matrix in the same direction (figure 5), the composite is transversely isotropic macroscopically; similarly, the periodic RVE is also transversely isotropic microscopically. Therefore, the macroscopic mechanical properties of the composite can be determined through research on RVE.  Assuming that the interfacial bonding between the CNTs and the alumina matrix is perfect in the entire loading process, and the effect of the stress transfer efficiency in the CNTs ends is not considered, the stress transfer theoretical model can be established ( figure 6). The length of the CNTs was 2L, and the Inner and Outer radii were R i and R o , respectively. The matrix thickness of the RVE is b. The effects of the stress transfer equation from the CNTs to the matrix are considered, and the interfacial stress distribution and length of CNTs on the load transfer are investigated separately. Using the above differential equation, the minimum length (dy) of CNTs is taken out in the y direction, and the distribution of shear stress on CNTs can be obtained through the axial normal stress change along the y-axis.
Because the thickness of CNTs is on the nanoscale, it can be assumed as t i = t ; in then,

Stress distribution of CNTs wall
The variation in stress along the load direction (y-direction) is considered by means of a classic shear-lag model, while the variation in stress perpendicular to the load direction (x direction) is ignored. Based on this assumption, the stress in the y-direction can be approximated as a function.
The relationship between t m and t i can be got from the above two equations. When the loads are along the y direction, t m can also be approximated by using the y-direction displacement.  (8) and (9), respectively. The shear stress t i is obtained as: Substituting equation (10) into equation (8), the shear stress of matrix can be obtained as follows: The displacement of matrix can be obtained from equations (9) and (11): The stress of the matrix along the y direction can be obtained from the above differential equation.
where s m denotes the axial stress applied to the matrix.
The stress equilibrium between the stress distribution and the stress distribution in the composite materials is as follows: The following boundary conditions are required for the stress equilibrium between the stress load s o exerted the composite and the stress distribution inside the composite: From equations (13) and (14), we can obtain Using equations (3), (10), and (15) simultaneously, we obtain Since the interfacial bonding is perfect, the interfacial strain is equal, and From equations (16) and (17), we can have Substituting into equation (3), the ordinary differential equation can be solved as Substituting A and B into equations (20) and (21), we can obtain

Effective length of CNTs
CNTs are embedded in the alumina matrix, and the axial normal stress may be increased from both ends of the CNTs to the middle section, gradually reaching the maximum value. Therefore, there exists an effective length L eff of CNTs in CNTs/alumina composite, the stress transfer efficiency of CNTs reinforced alumina composites is defined by L eff ; and L eff is the minimum length of CNTs when its stress reaches the maximum. When the length of CNTs is longer than the effective length L eff , CNTs cannot achieve better enhancement and may easily lead to collapse and folding. To solve the saturation stress, we can assume that the length L of CNTs is infinite; then,

Analysis of numerical results
Assuming that the interfacial bonding is perfect, as shown in figure 7, the interfacial stress distribution of the RVE in the CNT/alumina ceramic composites was calculated. s t is the tensile stress of the CNTs and t is the interfacial shear stress along the longitudinal axis of the nanotubes.
where e is the axial strain applied on the nanotube surface, L is the length of the CNTs (G m. is the shear modulus of the alumina matrix, E t is the elastic modulus of the nanotubes, and R is the radius of the RVE determined by the volume fraction of the nanotubes. where is the cross-sectional area, r 1 and r 2 are the inner and outer radii of the nanotubes, respectively. We take A t = 2.912 nm 2 , d = 3 nm, r 1 = 1.16 nm and r 2 = 1.5 nm.
The parameters needed for the calculation are E t = 1.05 TPa, G m = 143.44 GPa, e = 0.0686%, which is the maximum strain of the alumina matrix. R/r 2 = 7.236 5% is the volume fraction of the nanotubes in the composites. t = r 2 -r 1 is the wall thickness of the nanotubes, and is taken as 0.34 nm. From figure 8, it can be seen that the stress varies with the change in the CNTs' length; the stress has a maximum value and appears in the middle part of the nanotubes. When the length of the nanotubes was greater than 30 nm, the stress reached a maximum value. The maximum value did not change with an increase in the length of the nanotubes. If the interfacial bonding is perfect, fracture first appears in the middle of the CNTs. Figure 9 shows that the shear stress of the CNT/alumina interface is concentrated near the end of the CNTs and reaches a maximum at the end. When the strain of the alumina reached its maximum, the shear stress at the end of the CNTs was approximately 0.06 GPa. Moreover, the distribution of the shear stress at the end was not affected by the length of the nanotubes.
To prove the correctness of the theoretical derivation, the FEM software of ABAQUS was used to calculate a numerical example. The carbon nanotubes are divided by reduced-integration elements of C2D8R and alumina matrix is divided by quadrilateral elements of C2D10, and the number of elements is 25313. The boundary condition is fixed on the left, and free on the right. Figure 10 shows the calculation results of the FEM numerical results and the theoretical results. The theoretical maximum value of the normal stress of the CNTs was 626 MPa, the FEM numerical result was 576.8 MPa, and the error was 8.53% ( figure 10(a)). The theoretical maximum value of the interfacial shear stress was 108 MPa, and the closer the shear stress gradient to the nanotubes, the maximum value of the finite element simulation result was 77.9 MPa ( figure 10(b)). By comparing the stress distribution law, we found that the changing trends between the theoretical and numerical values were in good agreement with each other.

Conclusions
In this study, the interfacial stress transfer behavior of CNT-reinforced alumina ceramic composites was analyzed, and the stress distribution was simulated under the condition of perfect interfacial bonding. When the length of the nanotubes is longer than 30 nm, there is a maximum (0.68 GPa) normal stress in the middle of the CNTs. The maximum value did not change with an increase in the length of the nanotubes. At the ends of the CNTs, the stress gradient becomes larger, and the stress decreases rapidly, with a maximum (approximately 0.06 GPa) shear stress at the ends of the CNTs. Finally, to prove the correctness of the theoretical derivation, the theoretical results were compared with the FEM results, and the results between the theoretical and numerical values were in good agreement with each other.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Author contributions
Methodology, Y Y; Software, W G-F; Formal Analysis, Y Y; Writing-Original Draft, W G-F; Validation, W G-F; Writing-Review-Editing, W G-F.

Funding
This study received no external funding.

Institutional review board statement
Not applicable.

Informed consent statement
Not applicable.

Conflicts of interest
The authors declare no conflicts of interest.