# Fracture Toughness of Carbon Nanotube-Reinforced Metal- and Ceramic-Matrix Composites

Authors: Y. L. Chen; B. Liu; Y. Huang; K. C. Hwang
Journal: Journal of Nanomaterials
 (2011)

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## Abstract

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.


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## Body

## 1. Introduction

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 [1–10]. However, experimental and numerical studies show that the performance of such composites depends critically on the CNT/matrix interfacial characteristics  [11–13]. Interface strength and interface length are two of the most important factors that affect the mechanical properties of CNT-reinforced composites  and therefore have drawn the attention of many researchers.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 [14–21],  and researchers have tried in various ways, such as nonionic surfactant and ion bombardment [11, 16, 22, 23], to form covalent bonds between CNTs and the polymer matrix to strengthen the interface. In order to know whether longer CNTs and stronger interfaces definitely result in better mechanical properties of CNT-reinforced composites, Chen et al. [24]  studied the fracture toughness enhancement of CNT-reinforced polymer-matrix composites. They found that neither longer reinforcing CNTs nor stronger CNT/matrix interfaces can definitely lead to the better fracture toughness, and the optimal interfacial chemical bond density and the optimal CNT length are those making the failure mode just in the transition from CNT pull-out to CNT break.Meanwhile, the production and application of CNT-reinforced metal- and ceramic-matrix composites draw more and more attention. Ma and coworkers [25]  formed CNT-nano-silicon-carbide (SiC)/ceramic composites and reported a 10% improvement in the strength and fracture toughness as compared to the monolithic ceramics. These modest improvements are attributed to nanotube/matrix debonding and crack deflection. The techniques to form CNT/metal-oxide composites  as well as CNT/metal-matrix composites have been developed [26–31]. However, they did not provide the expected improvement in mechanical properties. Is Chen et al.’s conclusion [24]  still valid for these hard matrix composites? Is there some difference between the soft matrix and hard matrix CNT-reinforced composites? This paper will focus on the case with higher matrix stiffness and study the influence factors on the fracture toughness of CNT-reinforced composites.


## 2. Roadmap for Hierarchical Failure Analysis of CNT-Reinforced Composites

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 Figure1(a). This toughening effect of bridging CNTs is equivalent to that of nonlinear springs connecting the upper and lower crack surfaces, as shown in Figure 1(b). The force-displacement relation for these springs can be obtained by studying the pulling force F and pull-out displacement δ of a single CNT, as in Figure 1(c). This F-δ curve depends on the interfacial atomic bond properties, that is, on the interaction between atoms, as shown in Figure 1(d). Therefore, CNT-reinforced composites have three failure mode levels: a bond break at the atomistic level, CNT fiber failure mode at the mesoscopic level, and macroscopic crack propagation at the macroscopic level. To better understand and optimize the toughness of CNT-reinforced composites, this paper presents hierarchical failure analysis. We first adopt shear-lag theory to investigate CNT fiber in Section 3  and then use facture mechanics to study macroscopic-level failure in Section 4. Our conclusions are summarized in the last section.Figure 1
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.


## 3. Force-Displacement Relation of a Single CNT Pulled from the Matrix

There are a  large number of continuum mechanics studies on the fiber-reinforced composites, especially the widely  used shear-lag theory [32–37]. Chon and Sun [33]  studied stress distribution along a single reinforcing fiber of a randomly oriented chopped-fiber composite under the assumption of perfect bonding. Lawrence [34]  assumed the stiffness of the fiber is lower than that of the matrix  and investigated fiber pull-out from an elastic matrix. For ceramic composites, Marshall et al.   [35] studied the cracking in brittle  matrix. Hutchinson and Jensen [36]  treated debonding process as a mode 2 crack to study fiber debonding and pull-out, and based on these studies, Budiansky et al. [37]  accounted for an interfacial debonding resistance and studied the effects of debonding and initial stress on overall composite toughness. Many of these models are also applicable to CNT-reinforced composites. However, for the completeness and convenience to readers, the related analysis is still briefly presented in this section.


### 3.1. Shear-Lag-Model-Based Stress Analysis of the CNT and Matrix

According to the shear-lag theory, the interaction between the CNTs and the matrix that results from the chemical bonds shown in Figure2 is shear stress, which is related to the relative displacement between the CNT fiber and the matrix Δu. A bond break occurs when Δu reaches the critical shear displacement δb, which depends only on the type of functionalization bond at the interface, whereas the corresponding interface strength τb also depends on the interface bond density. The interface shear stress τ is assumed to be proportional to the relative displacement Δu, that is,
(1)τ(x)=kΔu(x)=k[um(x)-uf(x)],
where k=τb/δb is the shear stiffness of the interface  and um(x) and uf(x) are the axial or x-direction displacements of the matrix and the CNT fiber, respectively.Schematic diagram of shear-lag model for the interactions between the CNT and the matrix (a), and the geometric parameters (b).
(a)(b)Suppose the CNT and the matrix are both linear elastic, with  Young’s modulusEf and Em, respectively. A representative volume element (RVE) including a single CNT with embedded length L and diameter d, as shown in Figure 2, is adopted for analysis. With the balance conditions of the fiber and the matrix, the shear stress distribution can be derived [24]  (2)τ(x)=FτbCδb⋅1/(EmAm)cosh⁡(xCτb/δb)sinh⁡(LCτb/δb)+1/(EfAf)cosh⁡[(x-L)Cτb/δb]sinh⁡(LCτb/δb),
where Af and Am are the cross-section areas of the CNT and the matrix in the RVE, F is the pulling force, and C depends on the material constants and geometry parameters as(3)C=πd(1EfAf+1EmAm).
The distribution of the axial normal stress in the CNT can also be derived as(4)σ(x)=1Af[F-∫0xπdτ(x)dx].


### 3.2. Critical Pull-Out/Break Condition of CNTs

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.


#### 3.2.1. Critical Condition for CNT Fiber Break

Obviously, the maximum axial normal stress in the CNT is located at positionx=0 and can be expressed as(5)σ(x)|max⁡=σ(0)=FAf.
CNT break occurs when the maximum axial normal stress reaches CNT strength σfb, and the corresponding critical pulling force is(6)Fmax⁡σ=σfbAf.


#### 3.2.2. Critical Condition for Interfacial Debonding

According to   (2), the maximum shear stress on the interface may appear at either x=0 for  EfAf<EmAm  or  x=L for EfAf>EmAm. In CNT/polymer composites, the stiffness of the CNT is usually greater than that of the matrix, that is, EfAf>EmAm, as Chen et al. have previously discussed in [24], but in  metal- and ceramic-matrix composites   Young’s modulus of the matrix is close to that of the CNTs, so the stiffness of the CNT is less than that of the matrix, that is, EfAf<EmAm. The current paper is focused on this hard matrix regime  and sometimes also presents the results of soft matrix regime for comparison.(1)EfAf<EmAm Case (Hard Matrix)
From   (2), the maximum shear stress on the interface can be found at the x=0 position, and
(7)τ(x)|max⁡=τ(0)=FτbCδb⋅1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb).
The corresponding critical pulling force is
(8)Fmax⁡τ0=Cτbδbsinh⁡(LCτb/δb)1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb).
Together with    (6) and (8), the transition condition between the pull-out and break failure modes is
(9)σfbAfCτbδb1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb)>1(interfacialdebonding),σfbAfCτbδb1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb)<1(CNTfiberbreaking),
that is, a weak interface results in interfacial debonding and fiber pull-out, whereas a strong interface leads to fiber break.(2)EfAf>EmAm Case (Soft Matrix)
The most important difference from the hard matrix case (i.e.,EfAf<EmAm) is that the maximum shear stress on the interface is found at the x=L position, so the critical pulling force is [24]
(10)Fmax⁡τL=Cτbδbsinh⁡(LCτb/δb)1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf),
and the transition condition between the pull-out and break failure modes is [24]
(11)σfbAfCτbδb1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf)sinh⁡(LCτb/δb)>1interfacialdebonding,σfbAfCτbδb1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf)sinh⁡(LCτb/δb)<1CNTfiberbreaking.


### 3.3. Relation between Pulling Force and Pull-Out Displacement

For different possible failure modes, there are three types ofF-δ curves, as shown in Figures  3(a), 3(b), and 3(c). Here, F is the pulling force, and δ=Δu(0) is the pull-out displacement. The corresponding relations are given as below.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 whenEfAf<EmAm, and (c) interface failure with steady debonding when EfAf>EmAm.
(a)(b)(c)


#### 3.3.1. CNT Break Case

TheF-δ  relation can be obtained as (12)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δCσ,0,δ>δCσ,
where α=EmAm/(EfAf) is the stiffness ratio of the matrix to the CNT, and(13)δCσ=σfbδbCτb⋅cosh⁡(LCτb/δb)+1/αEfsinh⁡(LCτb/δb).


#### 3.3.2. Interface Debonding from the End of Pulling Force (Hard Matrix)

IfEfAf<EmAm, the interface begins to debond at the x=0 position where the relative displacement reaches δb. So the critical pull-out displacement for the interfacial debonding case is (14)δCτ0=δb.
At this time, the relative displacement at the x=L position is δb·(1/α)(1+(α2-1)/(αcosh(LCτb/δb)+1)), which is less than  δb, so the interface is not debonded completely  and can still stand some degree of shear stress. After this point, the F-δ curve decreases until the interface debonds completely. In the process of debonding, the CNT and the matrix can be analyzed as a similar system but with a shorter interface length L2 (L2<L). Both the pulling force F and the pull-out displacement δ are dependent on the new interface length L2 as follows:(15)F(L2)=Cτbδb⋅EfAfsinh⁡(L2Cτb/δb)cosh⁡(L2Cτb/δb)+1/α,δ(L2)=δb+(L-L2)(Cτbδb⋅sinh⁡(L2Cτb/δb)cosh⁡(L2Cτb/δb)+1/α).
Getting rid of L2 from the  above two equations, we can get the relation between the pulling force F and the pull-out displacement δ after the beginning of interface debonding as(16)δ=δb+{L-δbCτbln⁡[F+F2+α2(CτbδbEf2Af2-F2)α(EfAfCτbδb-F)]}×(FEfAf).
So the relation between the pulling force F and the relative displacement  δ  is(17)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δb,Fsoften(δ),δb<δ≤δmax⁡,
where Fsoften(δ) is solved from   (16)  and the maximum pull-out displacement δmax is determined by(18)∂δ(L2)∂L2=0.
This process is shown in Figure 3(b), and the critical point (δCτ0,Fmaxτ0) is given by   (14) and (8), respectively. It is interesting to  notice that after the initial debonding at   x=0, the interface first debonds gradually  and steadily, with the gradual deceasing pulling force, but when the pull-out displacement reaches δmax, the whole bonding interface debonds  simultaneously and the pulling force drops down to zero suddenly.


#### 3.3.3. Interface Debonding from the End Away from Pulling Force (Soft Matrix)

As discussed by Chen et al. in   [24], if   EfAf>EmAm, the F-δ  relation is (19)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δCτL,EmAmCτbδb(δb2-δ21-α2),δCτL<δ≤δb,
where(20)δCτL=δb⋅(α+1-α2cosh⁡(LCτb/δb)+α).TheF-δ  curve is shown in Figure 3(c), and the critical point (δCτL,FmaxτL) is given by   (20) and (10), respectively. Different from the other debonding case in Section 3.3.2, the interface debonds gradually and steadily, until it goes down to zero.


## 3.1. Shear-Lag-Model-Based Stress Analysis of the CNT and Matrix

According to the shear-lag theory, the interaction between the CNTs and the matrix that results from the chemical bonds shown in Figure2 is shear stress, which is related to the relative displacement between the CNT fiber and the matrix Δu. A bond break occurs when Δu reaches the critical shear displacement δb, which depends only on the type of functionalization bond at the interface, whereas the corresponding interface strength τb also depends on the interface bond density. The interface shear stress τ is assumed to be proportional to the relative displacement Δu, that is,
(1)τ(x)=kΔu(x)=k[um(x)-uf(x)],
where k=τb/δb is the shear stiffness of the interface  and um(x) and uf(x) are the axial or x-direction displacements of the matrix and the CNT fiber, respectively.Schematic diagram of shear-lag model for the interactions between the CNT and the matrix (a), and the geometric parameters (b).
(a)(b)Suppose the CNT and the matrix are both linear elastic, with  Young’s modulusEf and Em, respectively. A representative volume element (RVE) including a single CNT with embedded length L and diameter d, as shown in Figure 2, is adopted for analysis. With the balance conditions of the fiber and the matrix, the shear stress distribution can be derived [24]  (2)τ(x)=FτbCδb⋅1/(EmAm)cosh⁡(xCτb/δb)sinh⁡(LCτb/δb)+1/(EfAf)cosh⁡[(x-L)Cτb/δb]sinh⁡(LCτb/δb),
where Af and Am are the cross-section areas of the CNT and the matrix in the RVE, F is the pulling force, and C depends on the material constants and geometry parameters as(3)C=πd(1EfAf+1EmAm).
The distribution of the axial normal stress in the CNT can also be derived as(4)σ(x)=1Af[F-∫0xπdτ(x)dx].


## 3.2. Critical Pull-Out/Break Condition of CNTs

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.


### 3.2.1. Critical Condition for CNT Fiber Break

Obviously, the maximum axial normal stress in the CNT is located at positionx=0 and can be expressed as(5)σ(x)|max⁡=σ(0)=FAf.
CNT break occurs when the maximum axial normal stress reaches CNT strength σfb, and the corresponding critical pulling force is(6)Fmax⁡σ=σfbAf.


### 3.2.2. Critical Condition for Interfacial Debonding

According to   (2), the maximum shear stress on the interface may appear at either x=0 for  EfAf<EmAm  or  x=L for EfAf>EmAm. In CNT/polymer composites, the stiffness of the CNT is usually greater than that of the matrix, that is, EfAf>EmAm, as Chen et al. have previously discussed in [24], but in  metal- and ceramic-matrix composites   Young’s modulus of the matrix is close to that of the CNTs, so the stiffness of the CNT is less than that of the matrix, that is, EfAf<EmAm. The current paper is focused on this hard matrix regime  and sometimes also presents the results of soft matrix regime for comparison.(1)EfAf<EmAm Case (Hard Matrix)
From   (2), the maximum shear stress on the interface can be found at the x=0 position, and
(7)τ(x)|max⁡=τ(0)=FτbCδb⋅1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb).
The corresponding critical pulling force is
(8)Fmax⁡τ0=Cτbδbsinh⁡(LCτb/δb)1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb).
Together with    (6) and (8), the transition condition between the pull-out and break failure modes is
(9)σfbAfCτbδb1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb)>1(interfacialdebonding),σfbAfCτbδb1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb)<1(CNTfiberbreaking),
that is, a weak interface results in interfacial debonding and fiber pull-out, whereas a strong interface leads to fiber break.(2)EfAf>EmAm Case (Soft Matrix)
The most important difference from the hard matrix case (i.e.,EfAf<EmAm) is that the maximum shear stress on the interface is found at the x=L position, so the critical pulling force is [24]
(10)Fmax⁡τL=Cτbδbsinh⁡(LCτb/δb)1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf),
and the transition condition between the pull-out and break failure modes is [24]
(11)σfbAfCτbδb1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf)sinh⁡(LCτb/δb)>1interfacialdebonding,σfbAfCτbδb1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf)sinh⁡(LCτb/δb)<1CNTfiberbreaking.


## 3.2.1. Critical Condition for CNT Fiber Break

Obviously, the maximum axial normal stress in the CNT is located at positionx=0 and can be expressed as(5)σ(x)|max⁡=σ(0)=FAf.
CNT break occurs when the maximum axial normal stress reaches CNT strength σfb, and the corresponding critical pulling force is(6)Fmax⁡σ=σfbAf.


## 3.2.2. Critical Condition for Interfacial Debonding

According to   (2), the maximum shear stress on the interface may appear at either x=0 for  EfAf<EmAm  or  x=L for EfAf>EmAm. In CNT/polymer composites, the stiffness of the CNT is usually greater than that of the matrix, that is, EfAf>EmAm, as Chen et al. have previously discussed in [24], but in  metal- and ceramic-matrix composites   Young’s modulus of the matrix is close to that of the CNTs, so the stiffness of the CNT is less than that of the matrix, that is, EfAf<EmAm. The current paper is focused on this hard matrix regime  and sometimes also presents the results of soft matrix regime for comparison.(1)EfAf<EmAm Case (Hard Matrix)
From   (2), the maximum shear stress on the interface can be found at the x=0 position, and
(7)τ(x)|max⁡=τ(0)=FτbCδb⋅1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb).
The corresponding critical pulling force is
(8)Fmax⁡τ0=Cτbδbsinh⁡(LCτb/δb)1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb).
Together with    (6) and (8), the transition condition between the pull-out and break failure modes is
(9)σfbAfCτbδb1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb)>1(interfacialdebonding),σfbAfCτbδb1/(EmAm)+1/(EfAf)cosh⁡(LCτb/δb)sinh⁡(LCτb/δb)<1(CNTfiberbreaking),
that is, a weak interface results in interfacial debonding and fiber pull-out, whereas a strong interface leads to fiber break.(2)EfAf>EmAm Case (Soft Matrix)
The most important difference from the hard matrix case (i.e.,EfAf<EmAm) is that the maximum shear stress on the interface is found at the x=L position, so the critical pulling force is [24]
(10)Fmax⁡τL=Cτbδbsinh⁡(LCτb/δb)1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf),
and the transition condition between the pull-out and break failure modes is [24]
(11)σfbAfCτbδb1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf)sinh⁡(LCτb/δb)>1interfacialdebonding,σfbAfCτbδb1/(EmAm)cosh⁡(LCτb/δb)+1/(EfAf)sinh⁡(LCτb/δb)<1CNTfiberbreaking.


## 3.3. Relation between Pulling Force and Pull-Out Displacement

For different possible failure modes, there are three types ofF-δ curves, as shown in Figures  3(a), 3(b), and 3(c). Here, F is the pulling force, and δ=Δu(0) is the pull-out displacement. The corresponding relations are given as below.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 whenEfAf<EmAm, and (c) interface failure with steady debonding when EfAf>EmAm.
(a)(b)(c)


### 3.3.1. CNT Break Case

TheF-δ  relation can be obtained as (12)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δCσ,0,δ>δCσ,
where α=EmAm/(EfAf) is the stiffness ratio of the matrix to the CNT, and(13)δCσ=σfbδbCτb⋅cosh⁡(LCτb/δb)+1/αEfsinh⁡(LCτb/δb).


### 3.3.2. Interface Debonding from the End of Pulling Force (Hard Matrix)

IfEfAf<EmAm, the interface begins to debond at the x=0 position where the relative displacement reaches δb. So the critical pull-out displacement for the interfacial debonding case is (14)δCτ0=δb.
At this time, the relative displacement at the x=L position is δb·(1/α)(1+(α2-1)/(αcosh(LCτb/δb)+1)), which is less than  δb, so the interface is not debonded completely  and can still stand some degree of shear stress. After this point, the F-δ curve decreases until the interface debonds completely. In the process of debonding, the CNT and the matrix can be analyzed as a similar system but with a shorter interface length L2 (L2<L). Both the pulling force F and the pull-out displacement δ are dependent on the new interface length L2 as follows:(15)F(L2)=Cτbδb⋅EfAfsinh⁡(L2Cτb/δb)cosh⁡(L2Cτb/δb)+1/α,δ(L2)=δb+(L-L2)(Cτbδb⋅sinh⁡(L2Cτb/δb)cosh⁡(L2Cτb/δb)+1/α).
Getting rid of L2 from the  above two equations, we can get the relation between the pulling force F and the pull-out displacement δ after the beginning of interface debonding as(16)δ=δb+{L-δbCτbln⁡[F+F2+α2(CτbδbEf2Af2-F2)α(EfAfCτbδb-F)]}×(FEfAf).
So the relation between the pulling force F and the relative displacement  δ  is(17)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δb,Fsoften(δ),δb<δ≤δmax⁡,
where Fsoften(δ) is solved from   (16)  and the maximum pull-out displacement δmax is determined by(18)∂δ(L2)∂L2=0.
This process is shown in Figure 3(b), and the critical point (δCτ0,Fmaxτ0) is given by   (14) and (8), respectively. It is interesting to  notice that after the initial debonding at   x=0, the interface first debonds gradually  and steadily, with the gradual deceasing pulling force, but when the pull-out displacement reaches δmax, the whole bonding interface debonds  simultaneously and the pulling force drops down to zero suddenly.


### 3.3.3. Interface Debonding from the End Away from Pulling Force (Soft Matrix)

As discussed by Chen et al. in   [24], if   EfAf>EmAm, the F-δ  relation is (19)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δCτL,EmAmCτbδb(δb2-δ21-α2),δCτL<δ≤δb,
where(20)δCτL=δb⋅(α+1-α2cosh⁡(LCτb/δb)+α).TheF-δ  curve is shown in Figure 3(c), and the critical point (δCτL,FmaxτL) is given by   (20) and (10), respectively. Different from the other debonding case in Section 3.3.2, the interface debonds gradually and steadily, until it goes down to zero.


## 3.3.1. CNT Break Case

TheF-δ  relation can be obtained as (12)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δCσ,0,δ>δCσ,
where α=EmAm/(EfAf) is the stiffness ratio of the matrix to the CNT, and(13)δCσ=σfbδbCτb⋅cosh⁡(LCτb/δb)+1/αEfsinh⁡(LCτb/δb).


## 3.3.2. Interface Debonding from the End of Pulling Force (Hard Matrix)

IfEfAf<EmAm, the interface begins to debond at the x=0 position where the relative displacement reaches δb. So the critical pull-out displacement for the interfacial debonding case is (14)δCτ0=δb.
At this time, the relative displacement at the x=L position is δb·(1/α)(1+(α2-1)/(αcosh(LCτb/δb)+1)), which is less than  δb, so the interface is not debonded completely  and can still stand some degree of shear stress. After this point, the F-δ curve decreases until the interface debonds completely. In the process of debonding, the CNT and the matrix can be analyzed as a similar system but with a shorter interface length L2 (L2<L). Both the pulling force F and the pull-out displacement δ are dependent on the new interface length L2 as follows:(15)F(L2)=Cτbδb⋅EfAfsinh⁡(L2Cτb/δb)cosh⁡(L2Cτb/δb)+1/α,δ(L2)=δb+(L-L2)(Cτbδb⋅sinh⁡(L2Cτb/δb)cosh⁡(L2Cτb/δb)+1/α).
Getting rid of L2 from the  above two equations, we can get the relation between the pulling force F and the pull-out displacement δ after the beginning of interface debonding as(16)δ=δb+{L-δbCτbln⁡[F+F2+α2(CτbδbEf2Af2-F2)α(EfAfCτbδb-F)]}×(FEfAf).
So the relation between the pulling force F and the relative displacement  δ  is(17)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δb,Fsoften(δ),δb<δ≤δmax⁡,
where Fsoften(δ) is solved from   (16)  and the maximum pull-out displacement δmax is determined by(18)∂δ(L2)∂L2=0.
This process is shown in Figure 3(b), and the critical point (δCτ0,Fmaxτ0) is given by   (14) and (8), respectively. It is interesting to  notice that after the initial debonding at   x=0, the interface first debonds gradually  and steadily, with the gradual deceasing pulling force, but when the pull-out displacement reaches δmax, the whole bonding interface debonds  simultaneously and the pulling force drops down to zero suddenly.


## 3.3.3. Interface Debonding from the End Away from Pulling Force (Soft Matrix)

As discussed by Chen et al. in   [24], if   EfAf>EmAm, the F-δ  relation is (19)F={δCτbδb⋅EfAfsinh⁡(LCτb/δb)cosh⁡(LCτb/δb)+1/α,0≤δ≤δCτL,EmAmCτbδb(δb2-δ21-α2),δCτL<δ≤δb,
where(20)δCτL=δb⋅(α+1-α2cosh⁡(LCτb/δb)+α).TheF-δ  curve is shown in Figure 3(c), and the critical point (δCτL,FmaxτL) is given by   (20) and (10), respectively. Different from the other debonding case in Section 3.3.2, the interface debonds gradually and steadily, until it goes down to zero.


## 4. Macroscopic-Level Fracture Failure Analysis

### 4.1. Fracture Toughness EnhancementΔK

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 [38] (21)δ=2(1-νm2)KICEm2rπ=ηKICr,
where r is the distance to the crack tip  and η=22(1-νm2)/(Emπ) depends only on Young’s modulus Em and Poisson’s ratio νm. According to   (12), (17), and (19), for all different types of failure modes, the pulling force F(δ) can be expressed as a function of the distance to the macroscopic crack tip r, that is, F(δ)=F(δ(r))=F(r). The homogenized traction on the crack surface is then(22)p(r)=F(r)(Af+Am)-1,
and the fracture toughness enhancement ΔK can be computed as [39] (23)ΔK=∫0∞2p(r)πrdr.


### 4.2. Factors Affecting the Fracture Toughness EnhancementΔK

For the case of hard matrix (EfAf<EmAm), the factors affecting the fracture toughness enhancement ΔK are discussed below, and some results of soft matrix regime in   [24]  are also presented briefly for comparison.


#### 4.2.1. Effect of Interface LengthL

From the pull-out/break critical condition   (9), we know that if τb<(σfb)2/(CEf2δb), the failure mode should always be the CNT pull-out, as shown in Figure 4(a). In this case, the fracture toughness enhancement ΔK increases with an increase in the interface length L infinitely. This phenomenon is different from the case of weak interface in the soft matrix CNT-reinforced composites shown in Figure 4(b), in which an upper limit of the fracture toughness enhancement exists so any further lengthening of the CNTs improves the fracture toughness of the composites only slightly [24].Schematic diagram for the effect of the interface length on the fracture toughness enhancement: (a) hard matrix, weak interfaceτb<(σfb)2/(CEf2δb). (b) soft matrix, weak interface τb<(CσfbAf)2/(Em2Am2δb). (c) hard matrix, strong interface τb>(σfb)2/(CEf2δb); (d) soft matrix, strong interface τb>(CσfbAf)2/(Em2Am2δb).
(a)(b)(c)(d)For the case with strong interfaceτb>(σfb)2/(CEf2δb), as shown in Figure 4(c), a critical length LC exists and can be determined from the pull-out/break critical condition   (9):(24)LC=δbCτbln⁡σfb+Cτbδb(αEf)2-(σfb)2(α2-1)α(EfCτbδb-σfb).
When L<LC, the CNT is pulled out, and the fracture toughness enhancement ΔK increases with an increase in the interface length L. If L is further increased beyond the critical length  LC, the failure mode is converted from CNT pull-out to CNT break  and ΔK drops significantly. In this regime, the fracture toughness enhancement ΔK  decreases with an increase in L  and finally approaches the following value:(25)ΔKbreak|L→∞=2δbπCτb(σfb)2Af(Af+Am)ηKICEf.
It is interesting to note that, in this case, lengthening the interface beyond the critical length can even decrease the fracture toughness, which is very similar to the result shown is Figure 4(d) for the soft matrix CNT-reinforced composites [24].


#### 4.2.2. Effect of Interface Strengthτb

Another important factor affecting the toughness enhancement is interface strengthτb, and its effect is shown in Figure 5(a). When τb is small, the failure mode is CNT pull-out, and the fracture toughness enhancement ΔK increases with an increase in τb. Further increasing τb beyond the critical interface strength τbc, which can be determined by the pull-out/break critical condition   (9), leads the failure mode to be converted from CNT pull-out to CNT break, and ΔK drops significantly. In this regime, the toughness enhancement  ΔK  decreases with an increase in τb. This result agrees with the experimental and numerical studies of Xia et al.’s work on ceramic-matrix composites [40, 41], as well as the theoretical studies for soft matrix CNT-reinforced composites in Figure 5(b) [24]. Therefore, for CNTs with a given length, the maximum fracture toughness of the composite is achieved when τb   is only slightly smaller than the critical interface strength τbc.Schematic diagram for the effect of the interface strength on the fracture toughness enhancement: (a) hard matrix; (b) soft matrix.
(a)(b)


### 4.3. Optimization of Fracture Toughness Enhancement

In this subsection, we attempt to optimize the composite fracture toughness for the case ofEfAf<EmAm by tailoring both the interface length L and interface strength τb, which is essentially a bivariate optimization problem. For convenience, we use another group of variables, normalized interface length L̂=LCτb/δb and τb, instead. According to   (9), the maximum ΔK with a given L̂ can be achieved when(26)τb=τboptimal(L̂)=(σfbAfEmAm1+αcosh⁡(L̂)sinh⁡(L̂))2/(Cδb).
Figure 6 shows the normalized fracture toughness enhancement ΔK̂=ΔK·(Af+Am)ηKIC/σfbAfδb versus L̂ for cases with different stiffness ratios:  α=EmAm/(EfAf). It is found that when L̂→∞, ΔK reaches an infinite value. And according to   (26), the corresponding optimal interface strength is(27)τboptimal|L̂→∞=α(σfb)2AfπdδbEf(α+1).
This result is different from the case of EfAf>EmAm studied by Chen et al. [24], which presents the upper limit of the fracture toughness enhancement, as shown in Figure 6 with the dashed lines. When EfAf<EmAm, the value of the dimensionless fracture toughness enhancement ΔK̂ increases with L̂ unlimitedly, as shown in Figure 6 by the solid lines. So, there is neither upper limit for ΔK̂  nor optimal value for the CNT length L. The optimal CNT length L can only depend on the preparation level and some other factors such as the critical CNT length for clustering and self-folding [42].Figure 6
The normalized fracture toughness enhancementΔK̂ as a function of the normalized interface length L̂ with different stiffness ratio α=EmAm/(EfAf) and optimal interface strength.


## 4.1. Fracture Toughness EnhancementΔK

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 [38] (21)δ=2(1-νm2)KICEm2rπ=ηKICr,
where r is the distance to the crack tip  and η=22(1-νm2)/(Emπ) depends only on Young’s modulus Em and Poisson’s ratio νm. According to   (12), (17), and (19), for all different types of failure modes, the pulling force F(δ) can be expressed as a function of the distance to the macroscopic crack tip r, that is, F(δ)=F(δ(r))=F(r). The homogenized traction on the crack surface is then(22)p(r)=F(r)(Af+Am)-1,
and the fracture toughness enhancement ΔK can be computed as [39] (23)ΔK=∫0∞2p(r)πrdr.


## 4.2. Factors Affecting the Fracture Toughness EnhancementΔK

For the case of hard matrix (EfAf<EmAm), the factors affecting the fracture toughness enhancement ΔK are discussed below, and some results of soft matrix regime in   [24]  are also presented briefly for comparison.


### 4.2.1. Effect of Interface LengthL

From the pull-out/break critical condition   (9), we know that if τb<(σfb)2/(CEf2δb), the failure mode should always be the CNT pull-out, as shown in Figure 4(a). In this case, the fracture toughness enhancement ΔK increases with an increase in the interface length L infinitely. This phenomenon is different from the case of weak interface in the soft matrix CNT-reinforced composites shown in Figure 4(b), in which an upper limit of the fracture toughness enhancement exists so any further lengthening of the CNTs improves the fracture toughness of the composites only slightly [24].Schematic diagram for the effect of the interface length on the fracture toughness enhancement: (a) hard matrix, weak interfaceτb<(σfb)2/(CEf2δb). (b) soft matrix, weak interface τb<(CσfbAf)2/(Em2Am2δb). (c) hard matrix, strong interface τb>(σfb)2/(CEf2δb); (d) soft matrix, strong interface τb>(CσfbAf)2/(Em2Am2δb).
(a)(b)(c)(d)For the case with strong interfaceτb>(σfb)2/(CEf2δb), as shown in Figure 4(c), a critical length LC exists and can be determined from the pull-out/break critical condition   (9):(24)LC=δbCτbln⁡σfb+Cτbδb(αEf)2-(σfb)2(α2-1)α(EfCτbδb-σfb).
When L<LC, the CNT is pulled out, and the fracture toughness enhancement ΔK increases with an increase in the interface length L. If L is further increased beyond the critical length  LC, the failure mode is converted from CNT pull-out to CNT break  and ΔK drops significantly. In this regime, the fracture toughness enhancement ΔK  decreases with an increase in L  and finally approaches the following value:(25)ΔKbreak|L→∞=2δbπCτb(σfb)2Af(Af+Am)ηKICEf.
It is interesting to note that, in this case, lengthening the interface beyond the critical length can even decrease the fracture toughness, which is very similar to the result shown is Figure 4(d) for the soft matrix CNT-reinforced composites [24].


### 4.2.2. Effect of Interface Strengthτb

Another important factor affecting the toughness enhancement is interface strengthτb, and its effect is shown in Figure 5(a). When τb is small, the failure mode is CNT pull-out, and the fracture toughness enhancement ΔK increases with an increase in τb. Further increasing τb beyond the critical interface strength τbc, which can be determined by the pull-out/break critical condition   (9), leads the failure mode to be converted from CNT pull-out to CNT break, and ΔK drops significantly. In this regime, the toughness enhancement  ΔK  decreases with an increase in τb. This result agrees with the experimental and numerical studies of Xia et al.’s work on ceramic-matrix composites [40, 41], as well as the theoretical studies for soft matrix CNT-reinforced composites in Figure 5(b) [24]. Therefore, for CNTs with a given length, the maximum fracture toughness of the composite is achieved when τb   is only slightly smaller than the critical interface strength τbc.Schematic diagram for the effect of the interface strength on the fracture toughness enhancement: (a) hard matrix; (b) soft matrix.
(a)(b)


## 4.2.1. Effect of Interface LengthL

From the pull-out/break critical condition   (9), we know that if τb<(σfb)2/(CEf2δb), the failure mode should always be the CNT pull-out, as shown in Figure 4(a). In this case, the fracture toughness enhancement ΔK increases with an increase in the interface length L infinitely. This phenomenon is different from the case of weak interface in the soft matrix CNT-reinforced composites shown in Figure 4(b), in which an upper limit of the fracture toughness enhancement exists so any further lengthening of the CNTs improves the fracture toughness of the composites only slightly [24].Schematic diagram for the effect of the interface length on the fracture toughness enhancement: (a) hard matrix, weak interfaceτb<(σfb)2/(CEf2δb). (b) soft matrix, weak interface τb<(CσfbAf)2/(Em2Am2δb). (c) hard matrix, strong interface τb>(σfb)2/(CEf2δb); (d) soft matrix, strong interface τb>(CσfbAf)2/(Em2Am2δb).
(a)(b)(c)(d)For the case with strong interfaceτb>(σfb)2/(CEf2δb), as shown in Figure 4(c), a critical length LC exists and can be determined from the pull-out/break critical condition   (9):(24)LC=δbCτbln⁡σfb+Cτbδb(αEf)2-(σfb)2(α2-1)α(EfCτbδb-σfb).
When L<LC, the CNT is pulled out, and the fracture toughness enhancement ΔK increases with an increase in the interface length L. If L is further increased beyond the critical length  LC, the failure mode is converted from CNT pull-out to CNT break  and ΔK drops significantly. In this regime, the fracture toughness enhancement ΔK  decreases with an increase in L  and finally approaches the following value:(25)ΔKbreak|L→∞=2δbπCτb(σfb)2Af(Af+Am)ηKICEf.
It is interesting to note that, in this case, lengthening the interface beyond the critical length can even decrease the fracture toughness, which is very similar to the result shown is Figure 4(d) for the soft matrix CNT-reinforced composites [24].


## 4.2.2. Effect of Interface Strengthτb

Another important factor affecting the toughness enhancement is interface strengthτb, and its effect is shown in Figure 5(a). When τb is small, the failure mode is CNT pull-out, and the fracture toughness enhancement ΔK increases with an increase in τb. Further increasing τb beyond the critical interface strength τbc, which can be determined by the pull-out/break critical condition   (9), leads the failure mode to be converted from CNT pull-out to CNT break, and ΔK drops significantly. In this regime, the toughness enhancement  ΔK  decreases with an increase in τb. This result agrees with the experimental and numerical studies of Xia et al.’s work on ceramic-matrix composites [40, 41], as well as the theoretical studies for soft matrix CNT-reinforced composites in Figure 5(b) [24]. Therefore, for CNTs with a given length, the maximum fracture toughness of the composite is achieved when τb   is only slightly smaller than the critical interface strength τbc.Schematic diagram for the effect of the interface strength on the fracture toughness enhancement: (a) hard matrix; (b) soft matrix.
(a)(b)


## 4.3. Optimization of Fracture Toughness Enhancement

In this subsection, we attempt to optimize the composite fracture toughness for the case ofEfAf<EmAm by tailoring both the interface length L and interface strength τb, which is essentially a bivariate optimization problem. For convenience, we use another group of variables, normalized interface length L̂=LCτb/δb and τb, instead. According to   (9), the maximum ΔK with a given L̂ can be achieved when(26)τb=τboptimal(L̂)=(σfbAfEmAm1+αcosh⁡(L̂)sinh⁡(L̂))2/(Cδb).
Figure 6 shows the normalized fracture toughness enhancement ΔK̂=ΔK·(Af+Am)ηKIC/σfbAfδb versus L̂ for cases with different stiffness ratios:  α=EmAm/(EfAf). It is found that when L̂→∞, ΔK reaches an infinite value. And according to   (26), the corresponding optimal interface strength is(27)τboptimal|L̂→∞=α(σfb)2AfπdδbEf(α+1).
This result is different from the case of EfAf>EmAm studied by Chen et al. [24], which presents the upper limit of the fracture toughness enhancement, as shown in Figure 6 with the dashed lines. When EfAf<EmAm, the value of the dimensionless fracture toughness enhancement ΔK̂ increases with L̂ unlimitedly, as shown in Figure 6 by the solid lines. So, there is neither upper limit for ΔK̂  nor optimal value for the CNT length L. The optimal CNT length L can only depend on the preparation level and some other factors such as the critical CNT length for clustering and self-folding [42].Figure 6
The normalized fracture toughness enhancementΔK̂ as a function of the normalized interface length L̂ with different stiffness ratio α=EmAm/(EfAf) and optimal interface strength.


## 5. Conclusions

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 lengthL. However, for soft matrix composite, the fracture toughness improvement due to the reinforcing CNTs quickly becomes saturated with an increase in CNT length.It should be noted that the theoretical analysis and conclusions drawn in this paper can also be extended to fiber-reinforced composites.



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*Source: 746029-2010-12-12.xml*