# The Dependence of the Strength of a Carbon Fiber/Aluminum Matrix Composite on the Interface Shear Strength between the Matrix and Fiber

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Production of a Composite Carbon–Aluminum Wire

#### 2.2. Determination of Mechanical Properties

#### 2.3. Scanning Electron Microscopy and Elemental Analysis

#### 2.4. X-ray Diffraction Analysis

#### 2.5. Strength of the Interface between the Matrix and the Fiber

## 3. Results

#### 3.1. Microstructure

#### 3.2. Fiber Surface

#### 3.3. Strength and the Fracture Energy

## 4. Discussion

#### 4.1. Microstructure

#### 4.2. Fiber Surface

#### 4.3. Strength and the Fracture Energy

## 5. Numerical Calculations

_{cr}, the composite strength is determined by the strength of the fiber with a critical length, which increases with decreasing shear strength of the interface. This increase leads to a decrease in fiber strength according to the Weibull distribution and, as a result, in the composite strength (blue part of the curve σ* in Figure 10).

**, the composite strength is determined by the critical stress σ**

_{cr}**required for crack propagation, which, in contrast to σ*, decreases with the increasing shear strength of the interface (the red part of the curve σ**

_{C}**in Figure 10). This is because the shorter the critical length of the fiber, the less stress is required for crack propagation.**

_{C}_{cr}, which is determined by the intersection of the two above-mentioned curves σ

_{C}and σ*.

#### 5.1. Expression of the σ* Curve

_{f}is the strength of the fiber at the length l, σ

_{0}is the strength of the fiber at the length l

_{0}, m is the Weibull modulus, e is the base of the natural logarithm, and k is the refinement factor.

_{cr}from Equation (1) into Equation (2) as the fiber length gives an equation for the dependence of the fiber strength σ

_{f}on the shear strength τ:

_{0}and l

_{0}to be 4000 MPa over a length of 10 mm. To determine the coefficient k and the Weibull modulus m, we also used the manufacturer’s data, according to which a decrease in the fiber strength over a length of 25 mm does not exceed 5%. According to this, the strength of the fiber over a length of 25 mm was taken to be 3800 MPa. Thus, having two points with the coordinates σ

_{f}and l (4000;10) and (3800;25), the coefficients k and m were selected, which were 1.241 and 18, respectively. Figure 11 shows a plot of the dependence expressed by Equation (2) and the manufacturer’s data on the fiber strength.

_{f}is the fiber volume fraction and σ

_{m}is the strength of the matrix. Since aluminum and tin in the matrix alloy are insoluble in each other and do not form intermetallic compounds, the strength of the matrix could be calculated according to the mixture rule:

_{Al}is the volume fraction of aluminum in the matrix, σ

_{Al}is the strength of aluminum (80 MPa), and σ

_{Al}is the strength of tin (20 MPa). The matrix strength σ

**calculated by this method was between 66 and 73 MPa, depending on the tin content (see Figure 2b). Since this value was several orders of magnitude less than the fiber strength, for the convenience of further calculations, σ**

_{m}_{m}was taken to be 69 MPa.

#### 5.2. Expression of the σ_{c} Curve

_{E}is the elastic energy spent on the fracture of the composite wire, and A

_{Fract}is the work of fracture of the composite wire. The work of fracture is equal to the area under the curve in the load–displacement axes; its value for the wire in different states is shown in Figure 8. On the other hand, the work of fracture is expressed as the sum of several components:

_{PD}is the work expended in the plastic deformation of the matrix, A

_{NSm}is the work expended in the formation of new matrix surfaces, A

_{NSf}is the work expended in the formation of new fiber surfaces, and A

_{fr}is the work expended in overcoming the friction force between the matrix and the fiber.

_{A}. The work expended in overcoming the friction force is expressed as follows:

_{cr}/2, and the variable z is the change in the length of the unpulled part of the fiber in the range from l

_{cr}/2 to 0. After integrating and expressing the value of l

_{cr}through τ, according to Formula (1), Expression (9) takes the form:

_{cr}according to Equation (1), Equation (10) can be represented as:

_{cr}, calculated using Equation (11).

_{A}, which is the sum of the first three terms in Equation (8). In addition, since the component C

_{A}(0.98 mJ) is taken to be constant, the discrepancy between the remaining points should be the same.

**is the critical stress at which the crack propagates and the composite fails, E is the elastic modulus of the composite, and V**

_{c}_{E}is the volume of the material from which the stored elastic energy is transferred to the fracture work.

_{E}, the right side of Equation (12) was equated to the experimental values of the work of fracture, and the corresponding experimental values for the modulus of elasticity and the strength of the composite wire in different states were also used for E and σ

_{c}. The calculation showed that the value of V

_{E}almost did not change and was 0.33 ± 0.02 mm

^{3}. In this regard, for further calculations, the value of V

_{E}was assumed to be constant. Since the average value of the elastic modulus was 172 ± 10 GPa, for further calculations, this value was also assumed to be constant.

_{cr}= 107 MPa and σ

_{max}= 2675 MPa, respectively. The experimental values for the composite strength compared with the shear strength calculated from the experimental values of l

_{cr}are marked with black dots.

## 6. Conclusions

- Taking the example of a composite wire with an Al-25% Sn alloy matrix reinforced with carbon fiber, the dependence of the bending strength on the shear strength of the interface between the matrix and the fiber was determined. For this purpose, a series of composite wire specimens were obtained and subjected to heat treatment at temperatures from 300 to 600 °C. An analysis of the microstructure of the composite wire in an initial state and after heat treatment showed that there were almost no changes that could affect the strength of the composite.
- The examination of the surface of the carbon fiber extracted from the composite in an initial state and after heat treatment showed that the fiber in the initial state almost did not differ from the fiber in the delivered state. The fiber from the composite treated at 300 °C had a rougher surface. A further increase in the treatment temperature led to the formation of aluminum carbide crystals on the fiber surface, the size and number of which increased with increasing temperature.
- The study of the fracture surfaces and the comparison with the results of the investigation of the fiber surface showed that the emergence of carbide crystals at the interface led to a decrease in the length of the pulled-out part of the fiber. This explicitly indicated a relation with an increase in the strength of the interface. The evaluation of the work of fracture of the composite with different strengths of the interface between the matrix and the fiber demonstrated that as the strength of the interface increased, the work of fracture decreased, due to the premature fracture of the composite through crack propagation in one plane. In the case of weak interfaces, the trajectory of crack propagation is not in one plane, which leads to an increase in the fracture work.
- The effect of the shear strength of the interface between the matrix and the fiber on the strength, the modulus of elasticity, and the nature of the composite fracture was studied. The highest bending strength of 2450 MPa was observed for composite wire samples with the lowest interface shear strength. With an increase in the shear strength of the interface, a decrease in the composite wire strength below 900 MPa was observed. Along with the strength, as the shear strength of the interface increased, the flexural modulus also decreased, from 190 to 150 GPa.
- Based on the experimental data, as well as the refinement of the mixture rule according to the Weibull distribution and the estimation of the critical stress of crack propagation according to the Griffith–Orowan–Irwin concept, a numerical assessment of the dependence of the composite strength on the shear strength of the interface was made. On the basis of this, the critical shear strength was calculated, at which the greatest strength of the composite was achieved, the values being τ
_{cr}= 107 MPa and σ_{max}= 2675 MPa, respectively. It has been shown that the contribution of the work of overcoming the friction force to the total work of fracture at relatively small values of shear strength can be several times greater than the total contribution of all other types of energy. This indicates that a composite with weak interfaces can remain stable even after the onset of plastic deformation of the matrix and the breakage of some of the fibers in the bulk of the material.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Microstructure of the composite wire in an initial state (

**a**) and after heat treatment for 5 min at 500 °C (

**b**), 550 °C (

**c**), and 600 ° C (

**d**).

**Figure 2.**Change in the volume fraction of the carbon fiber (

**a**) and the tin structural component (

**b**) depending on the heat-treatment temperature.

**Figure 4.**Carbon fiber after the extraction from the matrix of the composite in an initial state (

**a**) and subjected to heat treatment at 300 °C (

**b**), 400 °C (

**c**), 500 °C (

**d**), 550 °C (

**e**), and 600 °C (

**f**). Images include inserts obtained at 50,000× magnification.

**Figure 5.**X-ray diffraction pattern of the fiber extracted from the composite after heat treatment at 600 °C.

**Figure 6.**Dependence of the strength and elastic modulus of the composite wire at three-point bending on the heat-treatment temperature.

**Figure 7.**Fracture surfaces of the composite wire in an initial state (

**a**) and after heat treatment at 500 °C (

**b**), 550 °C (

**c**), and 600 °C (

**d**).

**Figure 8.**Dependence of the length of the pulled-out fiber part and fracture work on the treatment temperature.

**Figure 9.**State diagram of the Al-Sn system [15].

**Figure 10.**Schematic representation of the dependence of the composite strength σ on the shear strength of the interface between the matrix and the fiber τ [4].

**Figure 11.**Plot of the fiber strength versus the length according to the Weibull distribution (red curve) and the manufacturer’s data on the fiber strength (black dots).

**Figure 12.**Experimental dependence of the work of fracture of the composite wire (blue line) and the calculated dependence of the work expended in overcoming the friction force (red line) on the critical length of the fiber.

**Figure 13.**Plots of the dependencies of the strength of the composite calculated according to the refined mixture rule (blue curve) and the critical stress of crack propagation in the composite (red curve). Experimental composite strength values are marked with black dots.

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**MDPI and ACS Style**

Galyshev, S.; Atanov, B. The Dependence of the Strength of a Carbon Fiber/Aluminum Matrix Composite on the Interface Shear Strength between the Matrix and Fiber. *Metals* **2022**, *12*, 1753.
https://doi.org/10.3390/met12101753

**AMA Style**

Galyshev S, Atanov B. The Dependence of the Strength of a Carbon Fiber/Aluminum Matrix Composite on the Interface Shear Strength between the Matrix and Fiber. *Metals*. 2022; 12(10):1753.
https://doi.org/10.3390/met12101753

**Chicago/Turabian Style**

Galyshev, Sergei, and Bulat Atanov. 2022. "The Dependence of the Strength of a Carbon Fiber/Aluminum Matrix Composite on the Interface Shear Strength between the Matrix and Fiber" *Metals* 12, no. 10: 1753.
https://doi.org/10.3390/met12101753