# The Influence of Fibre Cross Section Shape and Fibre Surface Roughness on Composite Micromechanics

## Abstract

**:**

## 1. Introduction

_{c}), which is of particular use when considering the strength of composite materials. From a micromechanics viewpoint, the concept of critical fibre length plays an important role in many of the single fibre experimental techniques used for characterising the stress transfer capability of the fibre–matrix interface [4]. In particular, the values of interfacial shear strength (IFSS) obtained from the single fragmentation test are related directly to the value of L

_{c}calculated from the experimental data [4,5,6].

## 2. Non-Circular Cross Section Glass Fibres

## 3. Critical Fibre Length

_{c}) involves balancing the peak tensile stress (σ

_{f}) transferred to a discontinuous fibre through the shear force (τ) at the interface [4,41,42,43]. For a fibre with cross sectional area (A

_{o}) and perimeter (P

_{o}), we obtain:

_{c}as

_{o}) with diameter (D), this has a cross sectional area A

_{f}= A

_{o}. It can be shown [42] that since a = Kb then the perimeter (P

_{f}) of the flat fibre is given by

_{c}from a circular fibre of the same cross sectional area may be followed. Unfortunately, there is no formula to calculate the exact perimeter of an ellipse (P

_{e}), but it can be approximated by

_{f}and C

_{e}parameters for fibres with various degrees of flatness parameter K. It can be seen that C

_{f}decreases as the flatness of the fibre cross section increases. Moreover, the values of C

_{f}and C

_{e}are less than unity for all values of K. Consequently, it appears that the critical fibre length for flat or elliptical cross section fibres is always smaller than that of a circular cross section fibre of equal area, strength and interfacial strength.

## 4. Composite Strength Effects

_{c}on the performance of these composites, and Figure 3 shows a summary of how the performance characteristics of composite modulus, strength, and notched impact in relation to the fibre length of a unidirectional discontinuous glass fibre-reinforced polypropylene [1,2,3,40]. Each of these properties has been normalised to the value predicted for continuous reinforcement length and so all values in Figure 3 fall in the range 0–1. It is important to note that, for the Kelly–Tyson form of critical fibre length, the fibre contribution to the composite strength has only reached 50% of the maximum fibre contribution at L

_{c}. To attain greater than 90% of the maximum theoretical strength would require using fibres with length >5 L

_{c}. In a similar vein, to attain 90% of the maximum notched impact in this system, it would require a fibre length >10 L

_{c}. Furthermore, the results shown in Figure 3 were generated for single values of fibre length; however, in real injection moulded SFT and LFT, there is usually a very broad range of fibre lengths present. Figure 4 shows some actual fibre length distributions obtained from injection moulded short glass fibre-reinforced polypropylene composites with different fibre contents [40]. It can be seen that a significant proportion of the fibres in such composites are shorter than L

_{c}and that the proportion of sub-critical fibres increase with increasing composite fibre content.

_{c}on composite strength can be modelled using micromechanical methods such as the Kelly–Tyson equation for the prediction of the strength (σ

_{uc}) of a composite reinforced with discrete aligned fibres [4,40,41,42,43]. This model is well known and can be expressed as σ

_{uc}= η

_{o}(X + Y) + Z, where Z is the matrix contribution, X is the sub-critical fibre contribution, and Y is the super critical contribution, in reference to the critical fibre length defined by Equation (1). Although the model was originally developed for aligned discontinuous fibre composites (η

_{o}= 1), it is often presented with an additional average fibre orientation factor (η

_{o}) when used for the performance of injection moulded composites.

_{c}(Equation (6) or (11)). It can easily be seen that if C is always less than unity for non-circular cross section fibres, then Equation (13) will always give a higher predicted value of stress than Equation (12).

_{f}= 1.8 GPa, and circular fibre diameter D = 13.6 µm [40,43]. It can be seen that the flat fibre composites have a consistently higher fibre contribution to their strength over the entire fibre length range. The higher strength contribution for flat fibres is also greater as the fibre content increases. The ratio of flat fibre strength contribution to circular fibre strength contribution is equal at all fibre contents, and the ratio of the two is shown in Figure 6 as a function of fibre length for different values of flatness (K). The flat fibre advantage, for the fibre contribution to composite strength, is a constant maximum for fibre lengths shorter than the flat critical fibre length (Equation (6)). Note that the flat critical fibre length decreases as K increases. The flat fibre strength advantage also increases in proportion to the value of K. For the typical K = 4 value of commercially available flat glass fibres, this means an apparent 33% greater contribution to composite strength for all fibres shorter that flat critical fibre length. For fibre lengths greater than the flat fibre L

_{cf}, the strength advantage decreases from the maximum value and tends to a value of unity as the fibres become much longer. This is understandable as the fibre strength contribution will tend towards that of continuous fibres, and this is assumed to be equal for all values of K in the analysis.

_{c}. With fibre lengths shorter than L

_{c}, they found that a thin wall ribbon shape filler, such as graphene, improved the reinforcement contribution to composite strength by a factor of two over that of a thin walled hollow filler of arbitrary shape. Similar to the results in Figure 6, they noted that this was due to the doubling of the interfacial area. For filler lengths greater than L

_{c}, they also noted that the maximum strength contribution, in both cases, tended towards the simple rule of mixtures for continuous reinforcement and, hence, the difference in the strength contributions of different shape fillers disappear (assuming they have the same ultimate strength), as also shown in Figure 6.

_{o}= 0.69 (typical for these injection moulded short fibre composites [43]) to the K-T model predictions. It can be seen that the flat fibre advantage is now dependent on the fibre content and decreases with decreasing fibre content. This is due to the fact that the matrix contribution to both flat and circular fibre-reinforced composites is equal and proportionally greater as the fibre content of the composites is decreased. Most interestingly, the data in Figure 7 now predicts that the flat fibre advantage is a maximum at the flat fibre L

_{cf}for any fibre content.

## 5. Natural Fibres

_{c}for an oval fibre can be as much as 25% lower than that of a circular cross section fibre of equivalent area in the K = 1 to 5 range. Nevertheless, it can also be noted in the images in Figure 11 that the irregular nature of the natural fibre surface means that the actual perimeter of these fibres is most likely still greater than that of a smooth ellipse. This would result in even larger potential reductions in L

_{c}for these fibres when compared to a circular cross section of equal area. Consequently, for those taking up the challenge of micromechanical modelling of natural fibres and their composites, it is clear that consideration must be given to the non-circular cross section of individual fibres when the L

_{c}parameter is being discussed. However, given the previously mentioned enormous range of inter-fibre and intra-fibre variation in natural fibre cross sectional shapes and areas, no further analysis is undertaken in this work.

## 6. Carbon Fibres

_{c}for that fibre type. Once again, it is beyond the scope of this single paper to discuss all of these fibre shapes, but it is noted that some of the fibres in Figure 12 will require a shape factor which will lead to significant reductions in actual L

_{c}values compared to the use of a circular cross section model. However, there is an additional aspect to the surface morphology of many carbon fibres, which does merit further discussion in terms of area:perimeter ratio and its effect on L

_{c}. Many carbon fibres have grooves distributed along the axial direction of fibres, which are inherited from the surface grooves of precursor fibres [45,46,47]. Researchers often use scanning electron microscopy or atomic force microscopy to visualise these grooved structures, as illustrated in Figure 13 [47]. The surface morphology, and, in particular, the dimensions of these surface grooves depend on the type of carbon fibre, the production process and conditions, and the post-production surface treatment. Sometimes, carbon fibres are designed to contain grooves to mechanically anchor the fibres to the resin, thereby minimizing fibre pull-out and improving composite interfacial performance [48]. However, given the preceding discussion, it is also possible that a contribution to any apparent improvement in interfacial adhesion may also be due to overlooked changes to the critical fibre length in the system due to an unaccounted for decrease in the fibre area:perimeter ratio, due to the surface roughness caused by these grooves.

_{rect}) of the fibre with a rectangular grooved surface is

_{rect}= A

_{o}, hence,

_{tri}) of the fibre with a triangular grooved surface is

_{tri}= A

_{o}, hence,

_{i}factors in Equations (15) and (17) are, therefore, the L

_{c}reduction factors, due to the increase perimeter of the fibres due to the surface grooves. Values for these factors are plotted in Figure 15 as a function of the value of b/a. Once again, it can be seen that consideration of the increased fibre perimeter, in this case due to surface grooves, may lead to significant changes in the value of critical fibre length for a carbon fibre depending on the dimensions of the grooves. Yao and Chen have published values of b/a up to 0.36 for T-300 PAN carbon fibres with different levels of surface etching [48]. This range is marked in Figure 15 and indicates a possible reduction of up to 30% in L

_{c}due to surface grooves. This groove related reduction would have to be subtracted in addition to any L

_{c}reduction due to non-circularity of the fibre cross section. This could result in a 50% or more error in characterising the critical fibre length of such fibres if considering them as approximately circular in cross section with a smooth surface.

## 7. Conclusions

_{c}) in composites with non-circular fibres is always less when compared to circular fibres with an equal cross-sectional area. Examples based on the shape of commercially available flat glass fibres revealed that L

_{c}depends on the flatness parameter of the fibres and can result in reductions of up to 30%. Further modelling of composite strength using the Kelly–Tyson theory predicted that flat fibres provide a consistently higher fibre contribution to composite strength over the entire fibre length range. This effect is at a maximum for short fibres with lengths below L

_{c}. Further modelling, which accounted for the matrix contribution and fibre orientation and length distribution effects typical in injection moulded composites, predicted a flat fibre advantage in composite tensile strength, which increased with fibre flatness and composite fibre content. This was shown to result in a predicted strength advantage for injected moulded short flat glass fibre-reinforced polypropylene of approximately 12–18% for typical moulding compound fibre contents in the 30–40% by weight range.

_{c}values compared to the use of a circular cross section approximation. The effects of surface roughness, in particular the grooved structure found on the surface of many carbon fibres, was also modelled. Results predicted a possible reduction of up to 30% in L

_{c}due to the surface grooves on carbon fibres. This groove-related reduction would have to be subtracted in addition to any L

_{c}reduction due to non-circularity of the fibre cross section, which could then result in 50% or more error in characterising the L

_{c}of such fibres if considering them as approximately circular in cross section with a smooth surface.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Prediction of normalised performance of GF-PP composite versus fibre length [4].

**Figure 4.**Some fibre length distributions in injection moulded GF-PP composites [40].

**Figure 5.**Kelly–Tyson equation prediction of flat and circular fibre strength contribution in unidirectional discontinuous GF-PP composites.

**Figure 6.**Kelly–Tyson equation prediction of flat/circular strength contribution ratio in unidirectional discontinuous GF-PP composites for fibre with different flatness ratios (K).

**Figure 7.**Kelly–Tyson equation prediction of flat/circle strength ratio for GF-PP including matrix contribution and average fibre orientation for injection moulded samples.

**Figure 8.**Kelly–Tyson equation prediction of stress–strain of injection moulded GF-PP with flat (K = 4) and circular cross-section fibres.

**Figure 9.**Prediction of flat/circle ratio of stress in injection moulded GF-PP composites for K = 4 flat fibres.

**Figure 10.**Prediction of flat/circle ratio of stress in injection moulded 40% GF-PP composites for different flatness fibres.

**Figure 12.**SEM of various carbon fibre cross sections (

**a**) T300, (

**b**) IM7, (

**c**) GT PAN, (

**d**) GT PAN/CNT-A (0.42 wt%), (

**e**) GT PAN/CNT-B1, (

**f**) GT PAN/CNT-A, and (

**g**) GT PAN/CNT-C (see reference [45] for further details).

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**MDPI and ACS Style**

Thomason, J.
The Influence of Fibre Cross Section Shape and Fibre Surface Roughness on Composite Micromechanics. *Micro* **2023**, *3*, 353-368.
https://doi.org/10.3390/micro3010024

**AMA Style**

Thomason J.
The Influence of Fibre Cross Section Shape and Fibre Surface Roughness on Composite Micromechanics. *Micro*. 2023; 3(1):353-368.
https://doi.org/10.3390/micro3010024

**Chicago/Turabian Style**

Thomason, James.
2023. "The Influence of Fibre Cross Section Shape and Fibre Surface Roughness on Composite Micromechanics" *Micro* 3, no. 1: 353-368.
https://doi.org/10.3390/micro3010024