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Hierarchical analysis of the fracture toughness enhancement of carbon nanotube- (CNT-) reinforced hard matrix composites is carried out on the basis of shear-lag theory and facture mechanics. It is found that stronger CNT/matrix interfaces cannot definitely lead to the better fracture toughness of these composites, and the optimal interfacial chemical bond density is that making the failure mode just in the transition from CNT pull-out to CNT break. For hard matrix composites, the fracture toughness of composites with weak interfaces can be improved effectively by increasing the CNT length. However, for soft matrix composite, the fracture toughness improvement due to the reinforcing CNTs quickly becomes saturated with an increase in CNT length. The proposed theoretical model is also applicable to short fiber-reinforced composites.

Carbon nanotubes (CNTs) possess exceptionally superior physical and mechanical properties, such as high strength, low density, high flexibility, and high toughness and therefore hold great promise for employment as reinforcements in advanced composites [

As a type of extraordinary reinforcements, CNTs can be incorporated in a polymer, metal, or ceramic matrix. The focus of many previous studies in CNT-reinforced composites has been on polymer-matrix materials [

Meanwhile, the production and application of CNT-reinforced metal- and ceramic-matrix composites draw more and more attention. Ma and coworkers [

In CNT-reinforced composites with macroscopic cracks, the high strength of CNTs can retard crack propagation, and a fracture zone bridged by CNTs at the crack tip is formed, as shown in Figure

Schematic diagram of three-level failure analysis models. (a) Fracture zone bridged with CNTs at the crack tip. (b) Macroscopic-level model with equivalent bridging nonlinear springs. (c) Mesoscopic-level model for studying CNT-fiber failure and obtaining the force-displacement relation of equivalent nonlinear spring. (d) Atomistic-level failure model for characterizing CNT/matrix interfacial bond breaking.

There are a large number of continuum mechanics studies on the fiber-reinforced composites, especially the widely used shear-lag theory [

According to the shear-lag theory, the interaction between the CNTs and the matrix that results from the chemical bonds shown in Figure

Schematic diagram of shear-lag model for the interactions between the CNT and the matrix (a), and the geometric parameters (b).

Suppose the CNT and the matrix are both linear elastic, with Young’s modulus

The two main fiber-level failure modes are usually interfacial debonding and fiber break, depending on the interfacial shear stress and the axial normal stress, respectively. When the composite is under increasing tension, both the shear stress on the interface and the axial normal stress in the CNT increase.

Obviously, the maximum axial normal stress in the CNT is located at position

According to (

From (

The most important difference from the hard matrix case (i.e.,

For different possible failure modes, there are three types of

The relation between the pulling force and the pull-out displacement for the three CNT-fiber failure modes (a) CNT breaking, (b) interface failure with partial steady debonding when

The

If

As discussed by Chen et al. in [

The

In CNT-reinforced composites, crack propagation is retarded by the pulling force of the CNTs at the crack surface, the so-called “bridge-toughening effect.” The displacement of the crack surface (i.e., half of the crack opening displacement) is [

For the case of hard matrix (

From the pull-out/break critical condition (

Schematic diagram for the effect of the interface length on the fracture toughness enhancement: (a) hard matrix, weak interface

For the case with strong interface

Another important factor affecting the toughness enhancement is interface strength

Schematic diagram for the effect of the interface strength on the fracture toughness enhancement: (a) hard matrix; (b) soft matrix.

In this subsection, we attempt to optimize the composite fracture toughness for the case of

The normalized fracture toughness enhancement

Combining the shear-lag model and fracture mechanics, we have carried out the hierarchical failure analysis on CNT-reinforced composites with hard matrix. The following conclusions have been reached.

(1) Stronger CNT/matrix interfaces cannot definitely lead to a better fracture toughness of these composites. In contrast, the optimal interfacial chemical bond density is that making the failure mode just in the transition from CNT pull-out to CNT break.

(2) For composites with hard matrix, there exists a critical interface strength, below which the CNT is always pulled out, and the fracture toughness can be effectively improved by increasing the interface length

It should be noted that the theoretical analysis and conclusions drawn in this paper can also be extended to fiber-reinforced composites.

The authors acknowledge the support of the National Natural Science Foundation of China (Grant nos. 10702034, 10732050, 90816006 and 10820101048) and the National Basic Research Program of China (973 Program), Grant nos. 2007CB936803 and 2010CB832701.