# Radial elastic heterogeneity in a high-modulus carbon fibre assessed by nano-indentation and micro-pillar compression

Vincent Keryvin, Guillaume Kermouche, Sergio Sao Joao

► To cite this version:

Vincent Keryvin, Guillaume Kermouche, Sergio Sao Joao. Radial elastic heterogeneity in a high-modulus carbon fibre assessed by nano-indentation and micro-pillar compression. Carbon, 2026, 247, pp.120991. 10.1016/j.carbon.2025.120991. emse-05385111

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https://hal-emse.ccsd.cnrs.fr/emse-05385111v1

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Research paper

# Radial elastic heterogeneity in a high-modulus carbon fibre assessed by nano-indentation and micro-pillar compression

V. Keryvin<sup>a,†</sup>,*, G. Kermouche<sup>b,†</sup>, S. Sao-Joao<sup>b</sup>

## ARTICLE INFO

Keywords:

Carbon fibre

Micro-pillar compression

Nano-indentation

Core/skin structure

Hardness

## ABSTRACT

Understanding the radial heterogeneity of carbon fibres is critical to optimising their performance in structural composites. In this study, we investigate the mechanical structure of a high-modulus PAN-based carbon fibre by combining instrumented nano-indentation (IIT) and micro-pillar compression (MCP). Nano-indentation mapping with 50 nm lateral resolution reveals a distinct layered skin-core structure. By combining IIT and MCP, we resolve for the first time the compressive modulus of individual radial layers within a single carbon fibre: a compliant core (110 GPa at 0.45% compressive strain), a transitional middle layer, and a stiff outer skin (520 ± 100 GPa, at 0.45% compressive strain). Our results confirm earlier indirect estimates and show that the skin modulus approaches that of highly oriented crystallites, whereas the core remains significantly softer due to misorientation. This integrated approach provides a robust framework for characterising anisotropic heterogeneous fibres and offers new insight into fibre-scale mechanical testing.

## 1. Introduction

High-modulus (HM) and ultra-high-modulus (UHM) carbon fibres derived from polyacrylonitrile (PAN) precursors represent a distinct subset within the broader family of continuous carbon fibres. Although they account for only a small fraction of composite applications (typically less than 10%), they are employed in structures where maximum stiffness is critical—such as racing yacht masts [1], deployable space structures [2], or high-performance sporting goods.

These fibres exhibit exceptionally high elastic moduli, typically around 400 GPa for HM fibres and 500 GPa for UHM fibres—around twice the stiffness of standard-modulus (SM) fibres. This superior performance arises from the near-perfect alignment of graphene sheets, the fundamental structural units of the carbon fibre microstructure, achieved during the final manufacturing stage: graphitisation [3]. This high-temperature process (2,000–3,000 °C) promotes crystallographic alignment and enhances longitudinal stiffness. However, it is generally accompanied by a reduction in tensile strength, restricting the use of HM and UHM fibres to stiffness-driven rather than strength-driven applications.

In longitudinal compression, continuous-fibre composites exhibit significantly lower strength than in tension [4], owing to instability mechanisms such as local buckling and kink-band formation. This behaviour is particularly critical in slender structures subjected to bending. While SM or intermediate-modulus (IM) fibre composites may

reach failure strains exceeding 1% [5,6], HM and UHM fibres typically fail at strains of ~0.5% or less [7–10]. Recent studies suggest that in PAN-based HM fibres, compressive failure in composites may be governed by the instability of the graphene layers themselves [4], as also proposed for pitch-based fibres [11]. The more perfectly oriented the graphene sheets—as in HM/UHM fibres—the lower the compressive strength appears to be. This microstructural anisotropy also affects the elastic response: both tensile and compressive tests have shown that the apparent modulus may evolve during loading [12], due to local reorientation of the graphene layers. In engineering composites design, this often leads to the adoption of compressive modulus values lower than their tensile counterparts, even when initial values appear similar [6].

New experimental techniques, such as micro-compression on micropillars (MCP) and instrumented indentation testing (IIT), now enable local assessment of the mechanical properties of individual fibres [13–17]. MCP studies have reported compressive moduli substantially lower than tensile ones, sometimes by a factor of nearly four [18]. IIT, on the other hand, yields an apparent modulus that differs from any true uniaxial modulus due to the strong anisotropy of carbon fibres, making interpretation non-trivial [14,19].

These discrepancies are likely exacerbated by the intrinsic microstructural heterogeneity of PAN-based fibres resulting from their

<footnote><sup>a</sup> Univ. Bretagne Sud, UMR CNRS 6027, IRDL, F-56321 Lorient, France</footnote>

<footnote><sup>b</sup> Mines Saint-Etienne, CNRS, UMR 5307 LGF, Centre SMS, F-42023 Saint-Etienne, France</footnote>

<footnote>* Corresponding author.
E-mail address: vincent.keryvin@univ-ubs.fr (V. Keryvin).</footnote>
manufacturing process especially the oxidation stabilisation stage [20], later exacerbated by the graphitisation stage for HM fibres [17]. Several authors have reported a resulting core-skin architecture [17,21, 22]. For instance, Chen et al. [23] employed plasma thinning, resistivity measurements, and tensile testing to demonstrate a substantial drop in modulus at the fibre core, with contrasts as high as a factor of 10 between the centre and the periphery.

Such unexpectedly low modulus values have raised questions in the community. Nevertheless, reliable stiffness measurements remain essential—for validating microstructure-based models [24] or assessing manufacturing effects [4]. This microstructural heterogeneity further complicates the interpretation of local mechanical measurements: responses obtained by IIT or MCP strongly depend on the region probed, the probed volume, and the micropillar diameter.

In this context, several key questions remain:

* Do PAN-based HM fibres indeed exhibit a pronounced core-skin structure?

* To what extent do the mechanical properties of the core differ from those of the full fibre?

* How can instrumented indentation results be reliably interpreted in such heterogeneous materials?

To address these issues, we investigate a PAN-based HM fibre using two complementary micromechanical techniques. Firstly, we conduct a shallow-depth nanoindentation campaign to map modulus variations across the fibre cross-section while minimising the probed volume. Secondly, we perform micro-compression testing on pillars of various diameters to detect any potential core-skin effects.

## 2. Materials and methods

### 2.1. Materials

The carbon fibres used in this study were commercial HR40 fibres (Mitsubishi, Japan), derived from PAN precursor. These fibres exhibit a nominally circular cross-section with a typical diameter of approximately 6 µm. he fibres in question originate from a unidirectional epoxy matrix composite reinforced with continuous carbon fibres. A polished cross-section perpendicular to the fibre direction was imaged using scanning electron microscopy (SEM, JEOL JSM 6301 F, Japan). An example is shown in Fig. 1, from which image analysis yields a more accurate average fibre diameter of 6.5 µm. A clear electron contrast is observed within individual fibres, suggesting the presence of a core region distinct from an intermediate transition zone and an outer shell.

### 2.2. Methods

#### 2.2.1. Instrumented indentation testing (IIT)

High surface quality, essential for shallow indentation depths, was achieved by successive polishing using silicon carbide abrasive papers (grit sizes 1000, 2000, 4000), followed by diamond suspensions (3 µm and 1 µm particle sizes), and finished with colloidal silica (0.03 µm).

IIT experiments were conducted using a Hysitron TI 980 TriboIndent (Brüker, Germany) equipped with a modified Berkovich tip. Each target fibre was first located using the scanning probe microscopy (SPM) mode. On each selected fibre, seven square indentation grids (22 × 22 indents) were positioned across the fibre diameter. The grids were spaced by 1 µm intervals using piezo-scanner actuation. Within each grid, the individual indent spacing was approximately 50 nm. This lateral resolution is made it possible due to the complete elastic behaviour of carbon fibres indented parallely to the graphene layers [25,26]. As reported by Guruprasad et al. [19], HR40 fibres exhibit fully reversible mechanical behaviour under shallow indentations, with negligible residual imprint for depths up to 150 nm. At the low loads employed here, interaction between adjacent indents is therefore considered negligible.

Indentation grids were generated using the accelerated property mapping (XPM) technique. A total of 3388 indents were performed. Each indent involved a peak load $P_{max} = 200$ µN applied over 0.1 s, held for 0.1 s, and unloaded in 0.1 s (corresponding to around 3 indents per second). Mechanical properties—indentation hardness $H$ and reduced modulus $M_{IT}$—were extracted from the load-displacement ($P-h$) curves following the methodology of Oliver and Pharr [27]. The key steps are recalled below. The contact depth $h_c$, at which the indenter is assumed to be in effective contact with the material, is computed by Eq. (1):

$$h_c = h_{max} - \epsilon \frac{P_{max}}{S} \qquad (1)$$

where $h_{max}$ is the maximum displacement under $P_{max}$, $S$ is the contact stiffness at the onset of unloading, and $\epsilon$ is a geometry-dependent factor accounting for sink-in. The stiffness $S$ is defined by Eq. (2):

$$S = \frac{dP(h)}{dh}(h_{max}) \qquad (2)$$

The unloading curve $P(h)$ was fitted using the following power-law form Eq. (3):

$$P(h) = A \cdot (h - h_f)^m \qquad (3)$$

where $A$, $m$, and $h_f$ are fitting parameters.

Due to the unusual complete elastic recovery behaviour of carbon fibres [26], this power-law fit is valid only over a limited portion of the unloading curve. A fitting window between 90% and 95% of $P_{max}$ was selected to ensure accuracy. The coefficient $\epsilon$, typically 0.72–0.75 for conventional materials [27], was recalculated from the exponent $m$ using the analytical expression Eq. (4) [27]:

$$\epsilon(m) = m \cdot \left( 1 - \frac{2 \Gamma \left( \frac{m}{2(m-1)} \right)}{\sqrt{\pi} \Gamma \left( \frac{1}{2(m-1)} \right)} (m-1) \right) \qquad (4)$$

where $\Gamma$ is the Euler gamma function.

The indentation modulus $M_{IT}$ is then given by Eq. (5):

$$M_{IT} = \frac{\sqrt{\pi} S}{2\beta \sqrt{A_c(h_c)}} \qquad (5)$$

Here, $\beta$ accounts for deviations from axisymmetry in the indenter geometry (taken as 1.034 for the modified Berkovich tip [27]), and $A_c(h_c)$ is the projected contact area, evaluated at the contact depth $h_c$ and calibrated on fused silica. The first term of the area function was fixed to the ideal value 24.5: $A_c(h_c) = 24.5 h_c^2 + \dots$

The true indentation modulus $M$ of the material is deduced by removing the contribution of the indenter (with Young’s modulus $E_i = 1, 100$ GPa and Poisson’s ratio $v_i = 0.07$), assuming a series spring model, as described by Eq. (6):

$$M = \left( \frac{1}{M_{IT}} - \frac{1 - v_i^2}{E_i} \right)^{-1} \qquad (6)$$

The indentation hardness $H$ is given by Eq. (7):

$$H = \frac{P_{max}}{A_c(h_c)} \qquad (7)$$

Using the spatial coordinates of each indent, high-resolution maps of modulus $M$ and hardness $H$ were generated for each fibre cross-section. Five different fibres were tested to assess repeatability and fibre-to-fibre variability.

#### 2.2.2. Micro-pillar compression testing (MCP)

The methodology follows the procedure described by Guruprasad et al. [18]. The key steps are briefly summarised below. Micro-pillars were fabricated by focused ion beam (FIB) milling using a Thermo Scientific Helios NanoLab DualBeam microscope. Initial coarse shaping was performed with a 30 kV, 2.5 nA beam, followed by final polishing
<figure><img src="image_3.png" /><figcaption>Fig. 1. Cross-section of HR40 carbon fibre polymer composite (SEM imaging) revealing electronic contrasts inside the fibre.</figcaption></figure>

at lower current (30 kV, 80 pA) to refine the geometry and achieve the desired dimensions. An example of a prepared pillars is shown in Fig. 2, with the flat punch indenter visible above. Three nominal pillar diameters were produced: 1.4, 3.4, and 5.3 µm. They are referred to hereafter as the 1, 3, and 5 µm pillars, respectively. For each size, two to three pillars were tested. Pillar characteristics are summarised in Table 1. These correspond to 23%, 57%, and 88% of the average fibre diameter, or equivalently 4.8%, 28%, and 66% of the fibre cross-sectional area. To mitigate buckling during compression, height-to-diameter ratios were kept below 3 in nearly all cases. All pillars were milled using an annular pattern from the outside inwards, minimising redeposition and ensuring approximate coaxiality with the parent fibre. The height and top/bottom diameters of each pillar were measured by SEM. The taper angle was consistently below 2°, and the pillars were nearly concentric with the fibre core, as illustrated in Fig. 3. In situ micro-compression experiments were performed using an Alemnis nanoindenter (Switzerland) mounted inside a Zeiss Gemini Supra 55VP SEM (Germany). A flat diamond punch with a 10 µm diameter was used to compress the pillars. Force-controlled loading was employed to enable precise probing of the elastic response without damaging the micro-pillars. This approach is preferable to displacement control in the early stages of loading, due to imperfect initial contact between the flat punch and the pillar top surface. For 1 µm pillars, maximum displacements remained within 20–30 nm, and the data were relatively noisy. A moving average filter was applied to the raw signal. Force (F) versus measured displacement (u<sub>mes</sub>) curves were recorded during the tests. Displacements were corrected for thermal drift and for the system’s elastic compliance (both frame and substrate contributions), yielding the corrected pillar displacement u<sub>pil</sub> via:

$$u_{\text{pil}} = u_{\text{mes}} - \frac{F}{K_{\text{eff}}}, \quad K_{\text{eff}} = \left( \frac{1}{K_f} + \frac{1}{K_s} \right)^{-1} \qquad (8)$$

Here, $K_f$ = 1.1 mN/nm is the stiffness of the loading frame. The substrate stiffness $K_s$, which accounts for deformation of the material beneath the pillar base, was estimated using the classical expression from Sneddon [28]:

$$K_s = 2 M_s r_{\text{bot}} \qquad (9)$$

where $M_s$ is the indentation modulus of the substrate, and $r_{\text{bot}}$ is the radius of the pillar base. This approach has been shown to provide accurate stiffness estimates in similar contexts [29,30]. Although the equipment is not aligned, meticulous handling has made it possible to work with different materials that undergo axisymmetric plastic deformation, providing high confidence in alignment. Engineering stress–strain curves were constructed from the force and corrected displacement data using the initial pillar height and cross-sectional area (evaluated at mid-height). In all cases, the curves exhibit a characteristic initial regime of increasing stiffness, attributed to progressive contact establishment between the imperfectly parallel top of the pillar and the flat punch surface [31]. This effect precludes a reliable determination of the compressive modulus at very small strains. To extract a representative compressive modulus $E_p$ for each pillar, the following analysis protocol was applied:

(a) identification of a linear portion of the stress–strain curve, bounded on one side by the end of contact establishment and on the other by the onset of material softening [12]; the extent of this linear regime varies with pillar size but remains consistent within each diameter group;

(b) computation of $E_p$ as the slope of the linear fit over this range;

(c) horizontal shifting of the entire curve so that the linear fit intersects the origin, allowing for a corrected zero-strain reference.
<figure><img src="image_4.png" /><figcaption>Fig. 2. HR40 carbon fibre micro-pillars of 3 µm in diameter after FIB machining (SEM imaging) and truncated conical punch indenter.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. HR40 carbon fibre micro-pillars of 1 µm in diameter after FIB machining (SEM imaging).</figcaption></figure>

# 3. Results

## 3.1. IIT

Figs. 4 and 5 present maps of indentation modulus and indentation hardness measured along a fibre diameter. The measurements are disturbed between two adjacent fibres, and the values in this region do not have physical meaning. Nevertheless, these contrasts reveal the fibre boundaries where polishing imperfections influence the results (this particular image was selected for this purpose). Both maps clearly indicate the presence of a softer, more compliant core and a stiffer, harder skin with a transitional zone between them. This is consistent

with the SEM image shown in Fig. 1. A profile of these two properties was extracted along a diameter, avoiding a polishing groove and edge regions that are less reliable for evaluation. These profiles are plotted in Figs. 6 and 7. Along the 6.5 µm fibre diameter, we can depict a three-zone structure by combining Figs. 1, 6 and 7. This conceptual representation is shown in Fig. 8. We observe a soft, compliant core approximately 2 µm in diameter, and a skin approximately 1.5 µm thick which is stiff and hard. Between these two zones, there is a transitional region more pronounced in the modulus (M) profile than in hardness (H).

This distribution enables estimation of the modulus $M_s$ used in Eq. (9) for analysing the MCP tests. Accordingly, we take values of 55
<figure><img src="image_6.png" /><figcaption>Fig. 4. Map of indentation modulus along the HR40 carbon fibre diameter.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Map of indentation hardness along the HR40 carbon fibre diameter.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Profile of indentation modulus along the HR40 carbon fibre diameter (line defined in Fig. 4).</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Profile of indentation hardness along the HR40 carbon fibre diameter (line defined in Fig. 5).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 8. Schematics of IIT results.</figcaption></figure>

**Table 1**

Geometrical characteristics of HR40 micro-pillars. The middle diameter (Ø) is the average diameter between the bottom and top diameters. 'Ratio' is the ratio height over middle diameter. 'Taper' is the angle between the lateral boundary of the pillar and the vertical. '% diameter' is the ratio middle Øover the fibre diameter in % and '% area' is the ratio cross-section area at middle Øover the cross-section area of the fibre.

<table><thead><tr><th>Micro-pillar #</th><th>Top Ø [µm]</th><th>Bottom Ø [µm]</th><th>Middle Ø [µm]</th><th>Height [µm]</th><th>Ratio [-] [°]</th><th>Taper [%]</th><th>%diameter [%]</th><th>%area [%]</th></tr></thead><tbody><tr><td>1-a</td><td>1.33</td><td>1.54</td><td>1.44</td><td>2.94</td><td>2.0</td><td>2.0</td><td></td><td></td></tr><tr><td>1-b</td><td>1.33</td><td>1.52</td><td>1.43</td><td>3.10</td><td>2.2</td><td>1.8</td><td></td><td></td></tr><tr><td>avg '1µm'</td><td>1.33</td><td>1.53</td><td>1.43</td><td>3.02</td><td>2.1</td><td>1.9</td><td>22</td><td>4.8</td></tr><tr><td>3-a</td><td>3.22</td><td>3.72</td><td>3.47</td><td>8.16</td><td>2.4</td><td>1.8</td><td></td><td></td></tr><tr><td>3-b</td><td>3.06</td><td>3.68</td><td>3.37</td><td>9.33</td><td>2.8</td><td>1.9</td><td></td><td></td></tr><tr><td>avg '3µm'</td><td>3.14</td><td>3.70</td><td>3.42</td><td>8.75</td><td>2.6</td><td>1.8</td><td>53</td><td>28</td></tr><tr><td>5-a</td><td>4.61</td><td>5.40</td><td>5.01</td><td>15.8</td><td>3.2</td><td>1.4</td><td></td><td></td></tr><tr><td>5-b</td><td>4.92</td><td>5.65</td><td>5.29</td><td>15.6</td><td>3.0</td><td>1.4</td><td></td><td></td></tr><tr><td>5-c</td><td>5.06</td><td>5.90</td><td>5.48</td><td>14.8</td><td>2.7</td><td>1.6</td><td></td><td></td></tr><tr><td>avg '5µm'</td><td>4.86</td><td>5.65</td><td>5.26</td><td>15.4</td><td>2.9</td><td>1.5</td><td>81</td><td>65</td></tr></tbody></table>

**Table 2**

Compressive moduli of HR40 micro-pillars with different diameters.

<table><thead><tr><th>Micro-pillar #</th><th>E<sub>p</sub> [GPa]</th></tr></thead><tbody><tr><td>1-a</td><td>104</td></tr><tr><td>1-b</td><td>125</td></tr><tr><td>avg '1µm'</td><td>115 ± 11</td></tr><tr><td>3-a</td><td>103</td></tr><tr><td>3-b</td><td>104</td></tr><tr><td>avg '3µm'</td><td>104 ± 1</td></tr><tr><td>5-a</td><td>230</td></tr><tr><td>5-b</td><td>178</td></tr><tr><td>5-c</td><td>218</td></tr><tr><td>avg '5µm'</td><td>209 ± 22</td></tr></tbody></table>

GPa for 1 µm diameter pillars, 60 GPa for 3 µm pillars, and 65 GPa for 5 µm pillars.

## 3.2. MCP

Figs. 9–11 present results for pillars of 1 µm, 3 µm, and 5 µm diameters. The $M_s$ values mentioned hereinbefore, allow for calculation of $K_s$ values, following Eq. (9), of 0.084, 0.222 and 0.367 mN/nm for diameters of 1 µm, 3 µm, and 5 µm, respectively. The establishment of contact between the indenter and the pillar allows determination of a modulus over strains ranging approximately from 0.3% up to nearly 0.6%. For clarity, we will state that moduli were determined around 0.45% strain. Table 2 summarises the measured moduli and

Fig. 12 compares the stress–strain response of the pillars for different diameters.

It can be observed that the 1 µm and 3 µm pillars exhibit similar stiffness values (~110 GPa). Thus, there exists a homogeneous zone in terms of this property over roughly half the fibre diameter (or one quarter in terms of cross-sectional area). However, this zone is about 3.3 times more compliant than the entire fibre, when compared to its tensile modulus (~360 GPa [12]).

In contrast, the 5 µm diameter pillars have a modulus approximately 1.6 times as high (~210 GPa). This stiffness, although still very low relative to the fibre's tensile modulus, corresponds to 81% of the fibre diameter (66% of the cross-sectional area) and suggests that the region between 3 and 5 µm diameter is stiffer than the core region, with a modulus exceeding 210 GPa. Fig. 13 proposes a schematic summarising the pillar properties in relation to the whole fibre.

# 4. Discussion

## 4.1. Core values of compressive modulus with respect to literature

The compressive modulus values measured in the core of the fibres appear extremely low. This had already been reported in one of our previous studies [18] on the same fibres, for pillar diameters of around 3.5 µm. It should be noted that very few studies exist on these micro-compression tests applied to carbon fibres [11,13,16,32], and to our knowledge no other work has been published on the stiffness apart from the one mentioned above [18], and definitely none with pillars of different diameters.

Nanoindentation studies [14,15] suffer from two main drawbacks for estimating a compressive modulus. Firstly, the indentation modulus measured is not the true compressive modulus due to the strong anisotropy of carbon fibres. Secondly, it is possible by inverse analysis to estimate three elastic moduli including the longitudinal modulus, but except for one study [19], all works assume that the longitudinal modulus involved is the fibre tensile modulus. Therefore, these studies do not provide *de facto* a measurement of the longitudinal compressive modulus.

The study by Guruprasad et al. [19] instead considered this longitudinal modulus as unknown. For the same HR40 fibre, in the core region of the fibre, with indentation depths small enough to elastically probe only this core, values of 116 GPa were obtained.<fnref n="1" /> However, this relatively low longitudinal modulus raised many questions. Our current study confirms these results as shown in Table 3.

<footnote n="1">¹ There was a typo in Table 1 of Guruprasad et al. [19] but not in subsequent tables and texts</footnote>
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Analysis of micro-pillar compression results on ø1 µm pillars. (a) Identification of a linear portion of the stress–strain curve, bounded on one side by the end of contact establishment and on the other by the onset of material softening; the extent of this linear regime varies with pillar size but remains consistent within each diameter group; computation of compressive modulus $E_p^c$ as the slope of the linear fit over this range. (b) Horizontal shifting of the entire curve so that the linear fit intersects the origin, allowing for a corrected zero-strain reference.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Micro-pillar compression results on ø3 µm pillars, see legend of Fig. 9 for details.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Micro-pillar compression results on ø5 µm pillars, see legend of Fig. 9 for details.</figcaption></figure>

Hao et al. [33] thinned the diameter of SM T700S PAN carbon fibres using cold plasma treatment and performed tensile tests on the residual thinned fibre. They found tensile moduli as low as 75 GPa for the core (1.5 µm core for a 7 µm diameter fibre). Chen et al. [23] used the same protocol on other fibres including M40J (which is a HM carbon fibre relatively similar to HR40). They performed electrical resistivity

measurements and then applied a correlation factor, they demonstrated to be valid between the entire fibre tensile modulus and electrical resistivity, finding a tensile modulus of 125 GPa for approximately 30% of the fibre diameter. This is in excellent agreement with our core results of around 110 GPa, see Table 3.
<figure><img src="image_14.png" /><figcaption>Fig. 12. Superposition of stress-strain curves for the pillars of different diameters in the [0.3:0.5]% strain range.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Schematics of micro-pillar compression results (top) and inferred compressive moduli of the different zones (bottom): the core (light blue and light green zones), the intermediate shell (light orange) and the sheath (red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<caption>**Table 3**</caption>

Comparisons of the compressive longitudinal moduli of HR40 carbon fibre in different zones by micro-pillar compression (MCP) and instrumented indentation technique (IIT) from the literature.

<table><thead><tr><th>Technique</th><th>MCP</th><th>MCP</th><th>MCP</th><th>MCP [18]</th><th>IIT [19]</th></tr></thead><tbody><tr><td>Diameter [µm] or zone</td><td>1.4</td><td>3.4</td><td>5.3</td><td>3.5</td><td>core</td></tr><tr><td>Modulus [GPa]</td><td>115</td><td>104</td><td>209</td><td>110</td><td>116</td></tr></tbody></table>

## 4.2. Estimation of the compressive modulus of the different fibre layers

We now attempt to estimate the modulus of each layer within the carbon fibre. We assume the fibre has an axisymmetric layered structure and that the compressive modulus of the fibre is equal to its tensile modulus, an assumption supported by identification studies on composite materials [12,34,35]. This microstructure allows us to express that the compressive stiffness is distributed proportionally to the areas and moduli of each layer, as done by Kobayashi et al. [36].

<caption>**Table 4**</caption>

Compressive moduli of HR40 different layers. For layer 3 (skin), different scenarios accounting for propagation of experimental uncertainties are considered with three scenarios: average, softest and stiffest.

<table><thead><tr><th>Layer<br>#</th><th>Radius<br>[µm]</th><th>%area<br>[%]</th><th>Modulus<br>[GPa]</th><th>Orientation angle<br>[°]</th></tr></thead><tbody><tr><td>1 (core)</td><td>0-1.7</td><td>28</td><td>110</td><td>29.6</td></tr><tr><td>2 (intermediate)</td><td>1.7-2.65</td><td>38</td><td>284</td><td>13.5</td></tr><tr><td>3 (skin - average)</td><td>2.65-3.75</td><td>34</td><td>523</td><td>6.2</td></tr><tr><td>3 (skin - softest)</td><td>2.65-3.75</td><td>34</td><td>424</td><td>8.7</td></tr><tr><td>3 (skin - stiffest)</td><td>2.65-3.75</td><td>34</td><td>622</td><td>3.8</td></tr></tbody></table>

Assuming a structure composed of two layers, 1 and 2, with areas $A_1$ and $A_2$, and total area $A$, we have:

$$E \cdot A = E_1 \cdot A_1 + E_2 \cdot A_2 \quad (10)$$

where $E$, $E_1$ and $E_2$ are the moduli of the whole structure and the two layers, respectively. Considering layer 1 as Pillar 3 µm and Pillar 5 µm
<figure><img src="image_16.png" /><figcaption>Fig. 14. Schematics of the procedure used to infer the compressive modulus of the carbon fibre skin.</figcaption></figure>

as the structure, the above relation allows calculating the modulus of layer 2, between the dimensions of Pillars 5 µm and 3 µm.

Using this modelling, we analyse the data with the following sequential procedure (illustrated in Fig. 14) for layers 2 (Pillar 5 µm) and 3 (sheath) as drawn in Fig. 13:

1. we take the initial value (zero strain) of the tensile modulus of the fibre from Ref. [12], $E_T^{f,0}$, with a known uncertainty of around 6% (358 ± 22 GPa).

2. we assume, as demonstrated in Ref. [12], that the initial compressive modulus of the fibre, $E_C^{f,0}$, is equal to the tensile one, $E_C^{f,0} = E_T^{f,0}$.

3. we consider the elastic softening from Ref. [6] that occurs during compressive deformation and evaluating the compressive modulus of the fibre at $\epsilon$ = -0.45% strain, for which the compressive modulus of the micro-pillars was determined. This is made using Eq. (11), introducing $\alpha'_f$ the linear decrease of the compressive modulus with compressive strain ($\alpha'_f \sim 23$ for HR40 carbon fibre [8]):

$$E_C^f(\epsilon) = E_C^{f,0}(1 + \alpha'_f \times \epsilon) \quad (11)$$

This gives $E_C^f(-0.45\%) = 321 \pm 20$ GPa

4. we use the value of the compressive modulus of the micro-pillars $E_P$ with a diameter of 5.3 µm and an uncertainty of around 10% (see Table 2) (209 ± 22 GPa)

5. we deduce the compressive modulus of the skin at 0.45% compressive strain, using Eq. (10), considering the uncertainties in the experimental data.

This procedure yields a compressive modulus of 523 ±99 GPa at a compressive strain of 0.45%

Results are added to Table 4. Using these results, we can summarise the properties of the fibre zones. They are schematised in Fig. 13. The core has a radius of approximately 1.75 µm with a modulus of 110 GPa (light blue and light green zones), then an intermediate zone less than 1 µm thick with an average modulus of about 280 GPa (light orange zone), and finally a skin zone approximately 1 µm thick with a modulus of around 520 ± 100 GPa (red zone).

### 4.3. Relation with microstructure

Kobayashi et al. [36] combined synchrotron X-ray Diffraction (XRD) and micro-Raman spectroscopy (1 µm resolution) measurements to study SM and HM fibres. In HM fibres, they reported a contrast factor of 2.7 between the modulus of the skin of the fibre and that of the fibre itself. In our case, we find a skin zone with a contrast of 1.4 (520/360) with respect to the entire fibre. Using a contrast of 2.7 would give a skin modulus of 972 GPa, much higher than the perfectly oriented crystallites (700 GPa [3]).

The elastic modulus of a carbon fibre $E_f$ has been proposed [3] to depend on the tensile ($e_{cr}$) and shear ($g_{cr}$) moduli of the crystallites and their orientation relative to the fibre axis, as shown by Eq. (12):

$$E_f(\theta)^{-1} = e_{cr}^{-1} + g_{cr}^{-1} (Z_2(\theta) - Z_4(\theta)) \quad (12)$$

where $Z_2$ and $Z_4$ are the second- and fourth-order cosine moments of the crystallite orientation distribution $\theta$ relative to the fibre axis. As a first approximation, assuming a constant $\theta$ distribution, $Z_2 = \cos^2 \theta$ and $Z_4 = \cos^4 \theta$. The crystallite moduli from Ref. [3] are $e_{cr} = 700$ GPa and $g_{cr} = 24$ GPa.

We use this relation between average crystallite orientation and modulus to estimate the orientation angle of each layer. The first and third layers seem homogeneous with regard to our results. The second layer is not, and an indicative mean orientation will be calculated. Results are added to Table 4. The first layer has an orientation angle of about 30°, and the last between 4 and 9°, depending on the modulus considered.

Zhou et al. [37] examined thermal treatment effects on carbon fibres. A 1000 °C treatment revealed a core with approximately 35° orientation using Transmission Electron Microscopy and selected-area electron-diffraction (SAED) patterns. Increasing graphitisation temperature to 2300 °C and 2800 °C, producing HM or UHM fibres, the core size decreases with orientations of 17° and 10°, respectively, while the outer zone (preceded by a transition zone) has an orientation of 2.5–5°. These results are consistent with our mechanical property-based estimations concerning the skin and transition zones but not the core. Core size estimates [37] were around 2 and 1.75 µm (for fibres <7 µm diameter), in line with our IIT measurements (2 µm) but somewhat smaller than the MCP estimates (>3.4 µm).
## 4.4. Insights and limitations

The results of the studies by Guruprasad et al. [18,19] on the core of HR40 carbon fibres, which had previously raised questions due to the surprisingly low values obtained, are now confirmed. The findings of this present work are consistent and demonstrate that their measurements were not artefacts. These HM carbon fibres are clearly heterogeneous in terms of mechanical properties. Nanoindentation studies must be performed within a homogeneous zone — for instance, a depth of 150 nm [19] corresponds to a lateral probed zone of approximately 1 µm, which is compatible with the estimated core size. Otherwise the indentation modulus values will be influenced by the transition and skin regions.

The compressive modulus values estimated by IIT (and reverse analyses involving three elastic moduli) are now directly validated by micro-compression tests. This reinforces the validity of the method reported in Ref. [19] when properly applied, and shows that previous IIT studies, failing to account for fibre heterogeneity and core properties lower than that of the entire fibre, may have led to incorrect estimates. Performing IIT beforehand to determine the indentation modulus of a given zone of the fibre is required to calibrate and interpret MCP tests. It is not only useful but necessary. However, it must be done at the appropriate location within the fibre to reflect the true properties of the targeted zone.

The consistency between literature results and our layer-by-layer modulus estimates also reinforces the multi-layer model. The modulus values inferred for the core, transition, and skin zones align well with previous estimates from indirect techniques, and are physically compatible with the expected structural and orientation gradients across the fibre radius.

The initial stiffness of the pillar is not always captured at the very beginning of the compression test, possibly due to contact settling. This can lead to a slight underestimation of the modulus, especially considering the non-linear behaviour starting below 0.5% strain, which lowers the apparent compressive stiffness. Future work should consider identifying and isolating this non-linear elasticity to refine modulus estimation.

The aim of this study is to produce new experimental evidence to inform the dialogue on the characterisation of carbon fibre microstructure. Further specific work is required to gain insight into the fibre microstructure.

## 5. Conclusion

To investigate the heterogeneous elastic behaviour of a high-modulus commercial carbon fibre, we combined two complementary mechanical characterisation techniques. The first is instrumented nanoindentation at shallow depths, enabling high-resolution mapping (50 nm) of the indentation modulus and hardness. This revealed a clear skin/core structure with a 1 µm thick core, a 1.5 µm thick skin, and a transition region in between. The second is micro-pillar compression (MCP), performed on pillars with diameters ranging from 22% to 81% of the fibre diameter. Nanoindentation results were used to ensure that each MCP test targeted a specific radial zone, making it possible — for the first time — to measure compressive moduli for distinct regions within the fibre.

From these tests, we determined the modulus of each layer: approximately 110 GPa for the core, 520 ± 110 GPa for the skin, and an intermediate value for the transition region. This study confirms the existence of a strong radial gradient in compressive stiffness, reinforcing previous findings based on nanoindentation and resolving earlier uncertainties regarding the low stiffness of the core. The skin exhibits moduli approaching theoretical maxima for well-aligned crystallites, whereas the core appears significantly more misaligned.

Overall, the combined use of shallow-depth nanoindentation and diameter-resolved MCP provides a powerful and coherent approach for probing the mechanical heterogeneity of advanced carbon fibres, with direct implications for understanding and optimising their compressive behaviour in composite applications.

## CRediT authorship contribution statement

**V. Keryvin:** Writing – original draft, Validation, Supervision, Methodology, Investigation, Data curation, Conceptualization. **G. Kermouche:** Writing – review & editing, Validation, Funding acquisition. **S. Sao-Joao:** Writing – review & editing, Investigation.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

The authors would like to thank T. Loho (University of Auckland) for assistance with the nano-indentation tests and T. Allen (University of Auckland, CAMMD) for valuable discussions. We also acknowledge A. Magueresse and A. Bendaoued (Univ. Bretagne Sud, IRDL) for their experimental support. V.K. thanks Univ. Bretagne Sud for funding partially his sabbatical stay (CRCT) at the Univ. of Auckland during which this paper was written. The University of Auckland's Centre for Advanced Materials Manufacturing is gratefully acknowledged for this research stay.

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