The effect of nanostructure on the tensile modulus of carbon fibers

Abstract

Using the Eshelby equivalent inclusion theory and the Mori–Tanaka method, a new micromechanical model is proposed to predict the tensile modulus of carbon fibers by considering crystallites, amorphous components, and microvoids of the fiber structure. Factors that affect the tensile modulus included the aspect ratio of crystallites, the aspect ratio of microvoids, the volume fraction of crystallites, the volume fraction of microvoids, and the orientation degree of crystallites. To follow the dependence of the tensile modulus of the fibers on microstructure, thirty different types of polyacrylonitrile-based fibers were prepared. The aspect ratios and orientation degrees of crystallites were calculated directly by X-ray diffraction. The aspect ratios and volume fractions of microvoids were obtained by small-angle X-ray scattering. The average tensile modulus of amorphous was estimated by dealing with thirty types of PAN-based fibers. The volume fractions of crystallites were obtained by the micromechanical model. Some relationships are concluded: (1) the tensile modulus increased with increasing volume fractions of crystallites, aspect ratios of crystallites and microvoids, and orientation degree of crystallites; (2) the tensile modulus increased with decreasing volume fractions of microvoids.

Introduction

Carbon fibers have high specific tensile modulus and strength, and play an important role in civil engineering construction, aerospace field, automotive materials, athletic equipment, etc. To improve the properties of carbon fibers, many studies have been undertaken to see the relationships between the mechanical properties and structure. The structure and morphology of carbon fibers have been studied by Oberlin [1], Johnson [2, 3], and Donnet [4]. Various mechanical models were introduced. The uniform stress mechanical model and the mosaic model of carbon fibers, both consisting of aligned crystallites connected with each other, were introduced to explain quantitatively the association between the tensile modulus and the crystallite orientation. The uniform stress mechanical model is able to describe the behavior of graphitized carbon fibers and shows the significance of the axial shear modulus on the bulk carbon fibers tensile modulus [5, 6]. The mosaic mechanical model can explain the measured increases in the tensile modulus and the crystallite orientation dependence on tensile stress [7]. Because the carbon fiber has a unique microstructure which consists of carbon crystallite layers, crystallite disorder regions (amorphous phase), and needle-like microvoids [8], a series–parallel mechanical model which comprises crystalline and amorphous phase is introduced. This series–parallel model can evaluate the heterogeneous stress distribution in the carbon fibers [9]. More recently, a two-phase composite micromechanical model which took into account the properties of both the crystalline and the amorphous components in the fiber structure was established to predict the tensile modulus of polyacrylonitrile (PAN)-based carbon fibers [10].

However, the uniform stress mechanical model and the mosaic model are unable to predict the behavior of low modulus PAN-based carbon fibers because they are too simplified and ignore the disordered structure. All models [5–7, 9, 10] mentioned above paid no attention to the microvoids whose content was about 10–20 volume percent in carbon fibers [11]. We think that there must be a connection between the volume percent of microvoids and mechanical properties of the carbon fibers. In this study, we introduce a new three-phase composite micromechanical model to predict the tensile modulus of PAN-based carbon fibers by considering the crystallites, the amorphous components, and the microvoids in the fiber structure. The combination of Eshelby’s solution [12, 13] and Mori–Tanaka’s mean stress method [14] is used to calculate the elastic coefficients. The X-ray diffraction (XRD) and small-angle X-ray scattering (SAXS) methods are applied to measure the experimental data of crystallites and microvoids in carbon fibers. Some factors that have an influence on tensile modulus of carbon fibers upon the microstructure are discussed.

Theoretical model

In this section, we establish the micromechanical model that is composed of crystallites, amorphous components, and microvoids to predict the tensile modulus of the carbon fibers. The amorphous components are assumed to be an isotropic matrix. The crystallites and the microvoids are regarded as inclusions in the isotropic matrix.

The true shape of crystallites and the microvoids are particularly complex in carbon fibers. However, it is reasonable that the shape of the crystallites is equivalent to oblate spheroids with the carbon layer axis parallel to their long axes. Moreover, the shape of the microvoids is equivalent to a long rotational ellipsoid [8, 10]. See Fig. 1. The concept of representative volume element (RVE) V in the carbon fibers is introduced. We assume that the RVE is subjected to remote boundary conditions, e.g., see Ref. [15, 16]. We also assume that (1) the amorphous carbon has a Poisson’s ratio of 0.3 [10]; (2) the coefficients of the elastic tensor of the crystallites are the same as those of graphite [17]; (3) the shape of the crystallites is an oblate spheroid, and the aspect ratio of the oblate spheroid ω _c is given by $\omega_{\text{c}} = L_{\text{a}} / \, L_{\text{c}},$where L _a is the crystallite diameter and L _c is the crystallite thickness, both of them are measured by XRD; (4) the shape of the microvoids is a rotation ellipsoid, and the aspect ratio of the rotation spheroid ω _v is given by ω _v = L/a, where L is the half length of rotation axis of the microvoid and a is the radius length, both of them are determined by SAXS; (5) the total volume fraction of crystallite, amorphous carbon, and microvoid is equal to unity.

Fig. 1

a model of the carbon fiber, b the structure of (a), (c) the modeling of (b)

A global coordinate system O−x ₁ x ₂ x ₃ is established in carbon fibers to ensure that the axis x ₃coincides with the fiber axis. Two local coordinate systems $O\, - \, x^{\prime}_{1} x^{\prime}_{2} x^{\prime}_{3}$ and $O\, - \, x^{\prime\prime}_{1} x^{\prime\prime}_{2} x^{\prime\prime}_{3}$ in the crystallite and microvoid are also established, respectively. Let the axis $x^{\prime}_{3}$ coincide with the rotation axis of crystallite and the $x^{\prime\prime}_{3}$ coincide with the microvoid rotation axis.

The carbon fiber is subjected to a uniform applied stress σ ₀, by Eshelby equivalent inclusion theory [17], we derive that the relation within the crystallite $(O\, - \, x^{\prime}_{1} x^{\prime}_{2} x^{\prime}_{3})$ is described by

$$ C_{0} \left( {\varepsilon^{{0^{\prime}}} + \,\tilde{\varepsilon }^{\prime} + S_{1} \varepsilon^{{*^{\prime}}} - \varepsilon^{{*^{\prime}}} } \right) = C_{1} \left( {\varepsilon^{{0^{\prime}}} + \tilde{\varepsilon }^{\prime} + S_{1} \varepsilon^{{*^{\prime}}} } \right), $$

(1)

where C ₀ and C ₁ are the elastic coefficients of the amorphous carbon and the crystallite carbon in the fibers, S ₁ is the fourth-order Eshelby tensor of the crystallite inclusion which has an oblate spheroid shape, e.g., see S ₁ in [18]. $\varepsilon^{{0^{\prime}}}$ is the strain in the isotropic matrix having no crystallites and microvoids, $\tilde{\varepsilon }^{\prime}$ is the average perturbed strain that caused by interactions between crystallites and microvoids, $\varepsilon^{{*^{\prime}}}$ is the equivalent eigenstrain of the crystallite.

Similarly, we can derive that the relation in the microvoid $(O\, - \, x^{\prime\prime}_{1} x^{\prime\prime}_{2} x^{\prime\prime}_{3})$. The mentioned relation is

$$ C_{0} \left( {\varepsilon^{{0^{\prime\prime}}} + \tilde{\varepsilon }^{\prime\prime} + S_{2} \varepsilon^{{**^{\prime\prime}}} - \varepsilon^{{**^{\prime\prime}}} } \right)\, = \,0, $$

(2)

where S ₂ is the fourth-order Eshelby tensor of the microvoid inclusion which has a rotational ellipsoid shape, e.g., see S ₂ in [19], $\varepsilon^{{0^{\prime\prime}}}$ is the strain in the isotropic matrix having no crystallites and microvoids, $\tilde{\varepsilon }^{\prime\prime}$ is the average perturbed strain that caused by interactions between crystallites and microvoids, and $\varepsilon^{{**^{\prime\prime}}}$ is the equivalent eigenstrain of the microvoid.

From (1) and (2), we obtain the equivalent eigenstrain of the crystallite ɛ ^* and the equivalent eigenstrain of the microvoid ɛ ^** in the carbon fiber $(O\, - \, x^{\prime}_{1} x^{\prime}_{2} x^{\prime}_{3})$ by coordinate transformation. The expressions for ɛ ^* and ɛ ^** are

$$ \varepsilon^{*} {\kern 1pt} = \, - \left( {T_{1}^{ - 1} } \right)^{\text{T}} \left( {\Delta {\text{CS}}_{1} + C_{0} } \right)^{ - 1} \Delta {\text{CT}}_{1}^{\text{T}} \left( {\varepsilon^{0} + \tilde{\varepsilon }} \right), $$

(3)

$$ \varepsilon^{**} = - \left( {T_{2}^{ - 1} } \right)^{\text{T}} \left( {S_{2} - I} \right)^{ - 1} T_{2}^{\text{T}} \left( {\varepsilon^{0} + \tilde{\varepsilon }} \right), $$

(4)

where ΔC = C ₁ − C ₀, I is the unit matrix, T ₁ and T ₂ are transformation matrices between the two local coordinate systems and the global coordinate system, respectively. See T ₁ and T ₂ in the appendix. ɛ ⁰ is the strain in the isotropic matrix having no crystallites and microvoids, $\tilde{\varepsilon }$ is the average perturbed strain that caused by interactions between crystallites and microvoids.

We let 〈ɛ〉₁ and 〈ɛ〉₂ be the average strains in a crystallite and a microvoid, and we obtain

$$ \langle \varepsilon \rangle_{1} = \varepsilon^{0} + \tilde{\varepsilon } + \varepsilon^{1} ,\,\,\,\,\langle \varepsilon \rangle_{2} = \varepsilon^{0} + \tilde{\varepsilon } + \varepsilon^{2} $$

(5)

where $\varepsilon^{ 1} = \left( {T_{ 1}^{ - 1} } \right)^{\text{T}} S_{ 1} T_{ 1}^{\text{T}} \varepsilon^{*}$ and $\varepsilon^{ 2} = \left( {T_{ 2}^{ - 1} } \right)^{\text{T}} S_{ 2} T_{ 2}^{\text{T}} \varepsilon^{**}$ are the perturbed strains by the crystallite and the microvoid. Substituting (3) and (4) into (5) and let $\varepsilon^{0} + \tilde{\varepsilon } = \langle \varepsilon \rangle_{M}$, we have

$$ \langle \varepsilon \rangle_{1} = \left[ {I - \left( {T_{1}^{ - 1} } \right)^{\text{T}} S_{1} \left( {\Delta {\text{CS}}_{1} + C_{0} } \right)^{ - 1} \Delta {\text{CT}}_{1}^{\text{T}} } \right]\langle \varepsilon \rangle_{M} , $$

(6)

$$ \langle \varepsilon \rangle_{2} = \left[ {I - \left( {T_{2}^{ - 1} } \right)^{\text{T}} S_{2} \left( {S_{2} + I} \right)^{ - 1} T_{2}^{\text{T}} } \right]\langle \varepsilon \rangle_{M} , $$

(7)

Based on Mori–Tanaka method, we derive that the average strain field 〈ɛ〉 _V in V as follows:

$$ \langle \varepsilon \rangle_{V} = \frac{1}{V}\int\limits_{{V - V_{1} - V_{2} }} {\langle \varepsilon \rangle_{M} {\text{d}}V} \int\limits_{{}} {} + \frac{1}{V}\int\limits_{{V_{1} }} {\langle \varepsilon \rangle_{1} {\text{d}}V} + \frac{1}{V}\int\limits_{{V_{2} }} {\langle \varepsilon \rangle_{2} {\text{d}}V} , $$

(8)

where V ₁ is the sum of the volume of all crystallites in V, and V ₂ is the sum of the volume of all microvoids in V.

Substituting (6) and (7) into (8), and considering orientation distribution of crystallites and microvoids, we have

$$ \langle \varepsilon \rangle_{V} \, = \,\left[ {I - f_{1} \left\{ {\left( {T_{1}^{ - 1} } \right)A^{1} T_{1}^{\text{T}} } \right\}_{{1\,{\text{angle}}}} - f_{2} \left\{ {\left( {T_{2}^{ - 1} } \right)^{\text{T}} A^{2} T_{2}^{\text{T}} } \right\}_{{2\,{\text{angle}}}} } \right]\langle \varepsilon \rangle_{M} , $$

(9)

where $A^{1} = S_{1} \left( {\Delta {\text{CS}}_{1} + C_{0} } \right)^{ - 1} \Delta C,\,\,\,\,\,\,A^{2} = S_{2} \left( {S_{2} - I} \right)^{ - 1},$ f ₁ and f ₂ are the volume fractions of the crystallites and the microvoids, f ₀ = 1 − f ₁ − f ₂ is the volume fraction of amorphous, $\left\{ \bullet \right\}_{i\,angle} = \int\limits_{0}^{2\pi } {\int\limits_{0}^{\pi } { \bullet \eta_{i} } } (\theta )sin\theta {\text{d}}\theta {\text{d}}\varphi / 2\pi \int\limits_{0}^{\pi } {\eta_{i} (\theta )} \sin \theta {\text{d}}\theta ,\,\,\,\,i = 1,\,2,$ and η ₁(θ) and η ₂(θ) are the distribution densities of the crystallites and microvoids, respectively.

Similarly, the average stress field 〈σ〉 _V in the carbon fiber can be derived as

$$ \left\langle \sigma \right\rangle_{V} = \frac{1}{V}\int\limits_{{V - V_{1} - V_{2} }} {\langle \sigma \rangle_{M} {\text{d}}V} \, + \,\frac{1}{V}\int\limits_{{V_{1} }} {\left\langle \sigma \right\rangle_{1} {\text{d}}V} , $$

(10)

where 〈σ〉₁ is the average stress of a crystallite, and 〈σ〉 _M is the average stress of amorphous. The expressions are

$$ \left\langle \sigma \right\rangle_{1} = T_{1} C_{1} T_{1}^{\text{T}} (\varepsilon^{0} + \tilde{\varepsilon } + \varepsilon^{1} ) = T_{1} C_{1} T_{1}^{\text{T}} \langle \varepsilon \rangle_{1} , $$

(11)

$$ \left\langle \sigma \right\rangle_{M} = T_{1} C_{0} T_{1}^{\text{T}} (\varepsilon^{0} + \tilde{\varepsilon }) = T_{1} C_{0} T_{1}^{\text{T}} \left\langle \varepsilon \right\rangle_{M} = C_{0} \left\langle \varepsilon \right\rangle_{M} . $$

(12)

So

$$ \langle \sigma \rangle_{V} = \frac{1}{V}\int\limits_{{V - V_{1} - V_{2} }} {C_{0} \langle \varepsilon \rangle_{M} {\text{d}}V} + \frac{1}{V}\int\limits_{{V_{1} }} {T_{1} C_{1} T_{1}^{\text{T}} \left\langle \varepsilon \right\rangle_{1} {\text{d}}V} $$

(13)

by substituting (11) and (12) into (10) yields

Substituting (6) into (13), then

$$ \begin{aligned} \langle \sigma \rangle_{V} = & \left[ {f_{0} C_{0} + f_{1} \left\{ {T_{1} C_{1} \left[ {I - S_{1} \left( {\Delta {\text{CS}}_{1} + C_{0} } \right)^{ - 1} \Delta C} \right]T_{1}^{\text{T}} } \right\}_{1\,angle} } \right]\langle \varepsilon \rangle_{M} \\ = & \left[ {f_{0} C_{0} + f_{1} \left\{ {T_{1} C_{1} \left[ {I - A^{1} } \right]T_{1}^{\text{T}} } \right\}_{1\,angle} } \right]\langle \varepsilon \rangle_{M} \\ \end{aligned} $$

(14)

Note that $\langle \sigma \rangle_{V} = \bar{C}\langle \varepsilon \rangle_{V}$, we have

$$ \bar{C} = \left[ {f_{0} C_{0} + f_{1} \left\{ {T_{1} C_{1} \left[ {I - A^{1} } \right]T_{1}^{\text{T}} } \right\}_{1\,angle} } \right] \cdot \left[ {I - f_{1} \left\{ {(T_{1}^{ - 1} )^{\text{T}} A^{1} T_{1}^{\text{T}} } \right\}_{1\,angle} - f_{2} \left\{ {(T_{2}^{ - 1} )^{\text{T}} A^{2} T_{2}^{\text{T}} } \right\}_{2\,angle} } \right] $$

(15)

Experimental

Materials

In order to research the dependence of the mechanical properties of the carbon fibers on nanostructure, we prepared thirty different types of PAN-based carbon fibers, with Young’s moduli in the range 200–500 GPa. All experimental fibers were supplied by tuozhan fiber Co., Ltd and were measured by XRD and SAXS. Their properties are listed in Table 1.

Table 1 Physical and mechanical properties of the carbon fibers

XRD measurements

The XRD experiment was performed at National Laboratory of tuozhan fiber Co., Ltd. Weihai, China using CuKα radiation (λ = 0.1541 nm) under the operating conditions of 40 kV and 40 mA. We carried out axial scan, radial scan, and azimuth angle scan with 2θ-scan step 0.02° and 0.504°, respectively.

SAXS measurements

SAXS measurement was carried out at the beam line (BL16B1) of the Shanghai synchrotron radiation facility with a wavelength of 0.124 nm. The distance from sample to the detector was 5120 mm.

Results and discussion

XRD

L _c and L _a can be obtained from (002) diffraction peak of axial scan profile and (100) diffraction peak of radial scan profile using Scherrer’s formula

$$ L = K\lambda /B\cos \theta, $$

(16)

where θ is the scattering angle, λ is the wavelength of the X-rays used, and B is the full width at half maximum (FWHM) of diffraction peak. The form factor K is 0.9 for L _c and 1.84 for L _a [20]. The FWHM of an azimuthal scan through the (002) reflection H can be directly obtained from azimuth angle scan profile. The value of the average interlayer spacing d ₀₀₂ can be calculated using Bragg’s law

$$ d_{002} = n\lambda /2\sin \theta, $$

(17)

where n is an integer. The apparent crystallinity of the carbon fiber X _c is given by [21]

$$ X_{\text{c}} = S_{\text{c}} /S_{\text{t}} \, \times \,100\,\%, $$

(18)

where S _c is the area of (002) diffraction Peak and S _t is the total area under the diffraction curve in the interval [14°, 34°]. The degree of crystallites orientation Π can be calculated applying the formula

$$ \varPi = \left( {180^{ \circ } - H} \right)/180^{ \circ } \, \times \,100\,\% . $$

(19)

Detailed data are presented in Table 1.

Thirty different types of experimental carbon fibers were prepared to obtain the tensile modulus of amorphous, see Table 1. Figure 2 shows the bulk tensile modulus E _bulk versus their crystallite thickness L _c. The tensile modulus of fibers that are composed of amorphous carbon and microvoids E _a+m = 111.48 GPa. This value was obtained by extrapolating the straight linear line (linear regression) of E _bulk versus L _c to zero position [9]. To simplify the calculations, let the shape of microvoids in the carbon fibers degenerate the sphere-like. Substituting E _a+m into (15) and let the average microvoids volume fraction of thirty types carbon fibers replace f ₂, we obtain the tensile modulus of amorphous carbon E _a, the value is 137.98 GPa.

Fig. 2

Linear fiting plot of the data that reflect the relationship between the L _c and the tensile modulus for thirty different types of carbon fibers

SAXS

In order to obtain the radius of gyration R _g, the Guinier analysis is the most popular method in the SAXS analyses [22, 23]. According to Guinier theorem, we derive the expression of the intensity

$$ I(h) = I^{0} \exp \left( { - \frac{{h^{2} \, R_{\text{g}}^{2} }}{3}} \right), $$

(20)

where h = (4π sin θ)/λ, λ is the wavelength of X-ray, 2θ is the scattering angle, and I ⁰ is the scattered intensity when h = 0. We can estimate the R _g value of the microvoids from the slope of the Guinier plot lnI(h) versus h ². The Jellinek Successive tangents method is used here [24]. We divide the microvoids size into three classes according to the tangent line of the Guinier plot. We derive each class of the gyration radius R _gi , i = 1, 2, 3. The volume fraction of each class microvoids W _i , i = 1, 2, 3 and the average gyration radius $\overline{{R_{g} }}$ can be calculated respectively, by

$$ W_{i} = \frac{{K_{i} /R_{{gi}}^{3} }}{{\sum\nolimits_{i} {K_{i} /R_{{gi}}^{3} } }}\quad i = 1,2, $$

(21)

$$ {\overline{R_{g}} } = \sum\limits_{i} {R_{{gi}} W_{i} } \quad i = 1,2,3, $$

(22)

where K _i is the intercept of the ith class tangent line in the vertical axis of the Guinier plot.

The radius of gyration at the direction of equator R _gc is independent of the length of rotation ellipsoids. Applying the formula $a = \sqrt 2 R_{\text{gc}}$, we can calculate the radial radius of the microvoids a. The difference of the microvoids in size is very small at the direction of equator [25], so we do not need to divide the radius of gyration at the direction of equator R _gc into three classes. We only divide the radius of gyration at the direction of meridian R _gm. Substitute a into the formula

$$ R_{\text{gmi}} = \sqrt {\frac{{a^{2} }}{5} + \frac{{L_{i} }}{5}} \quad i = 1, 2, 3, $$

(23)

then L = ∑  _i W _i L _i $\quad i = 1, 2, 3.$ Detailed data are presented in Table 2.

Table 2 Experiment results of microvoids for the PAN-based carbon fibers

The volume fraction of microvoids f ₂ is given by the equation [26, 27]

$$ f_{2} \, = \,\frac{{2\pi m^{2} c^{4} }}{{e^{4} \lambda^{3} (\Delta \rho )^{2} Dt{\text{AI}}^{0} }}\int\limits_{0}^{\infty } {I_{a} (h)} h\,{\text{d}}h, $$

(24)

where I _a (h) is the absolute X-ray intensity, which is determined by the direct method [28], Δρ is the electron density difference between the voids and the solid, m is the electron mass, c is the velocity of light, e is the elementary electric charge, D is the sample-to-detector distance, t is the specimen thickness, and AI⁰ is the incident X-ray beam flux. See f ₂ in Table 1.

Calculated volume fractions of crystallites

To determine the volume fractions of crystallites of T series carbon fibers—T300, T700, and T800—and MJ series—M35J and M40J—the distribution density function of crystallites and microvoids must be first confirmed. In the case of carbon fibers with a high degree of crystallite orientation in the present study, the distribution density function η ₁(θ) of the orientation angle θ between the normal of the carbon layer and the fiber axis can be closely approximated by an orientation distribution function of the form [29]

$$ \eta_{1} (\theta ) = K\sin^{\omega } \theta, $$

(25)

where

$$ K = \frac{{\varGamma \left[ {\left( {\omega + 3} \right)/2} \right]}}{{2\pi^{3/2} \varGamma \left[ {\left( {\omega + 2} \right)/2} \right]}} $$

$$ \omega = - \ln 2/\ln \left[ {\sin \left( {\pi \varPi /2} \right)} \right] $$

Γ(x) is a gamma function having the following property Γ(x + 1) = xΓ(x) (real x > 0).

Assume that the rotation axis of crystallite is almost vertical to the rotation axis of microvoid. The orientation distribution function of microvoids is determined by the formula [30]

$$ \eta_{2} \left( \theta \right) = \frac{\omega + 1}{4\pi }\left| {\cos \theta^{{}} } \right|^{\omega }, $$

(26)

where $\omega = - {\text{ln2}}/ { \ln } \left[ { { \sin } \left( {\pi \varPi / 2} \right)} \right].$

The volume fractions of crystallites of five samples can be calculated by substituting (25) and (26) into Eq. (15). See Fig. 3. The volume fractions of crystallites of five samples have a trend along the growth direction of tensile modulus. This trend is the same as the trend of apparent crystallinity.

Fig. 3

Calculated volume fractions of crystallites

Influence factors of tensile modulus

The orientation degree of crystallites Π, aspect ratios of crystallites ω _c, aspect ratios of microvoids ω _v, volume fractions of crystallites f ₁, and volume fractions of microvoids f ₂ are calculated by Eq. (15). Figure 4a (horizontal axis represents the increment of a factor, the same below) shows that the tensile modulus increases with increasing Π. The high modulus of a carbon fiber stems from the fact that the carbon layers, though not necessarily flat, tend to be parallel to the fiber axis [31]. The greater the degree of alignment of the carbon layers parallel to the fiber axis, the greater the fiber’s tensile modulus. Comparing T series carbon fibers with MJ series carbon fibers, we can find that the MJ series have larger Π than the T series.’ Five curves in Fig. 4a are different in length, because the five orientation degrees of crystallites had different initial values, which lie in the range 79–88 %, and the limit of orientation degree is 100 %. With Π close to 100 %, increases of tensile modulus gradually become slow. In Fig. 4b, the tensile modulus is plotted against ω _c. It can be seen that the tensile modulus increases linearly with ω _c in the interval 0 % ≤ ω _c ≤ 24 %. In a carbon fiber, there can be graphite regions of size L _c perpendicular to the carbon layers (crystallites) and size L _a parallel to the carbon layers. Both L _c and L _a grow with increasing heat treatment temperature in the processing of production of carbon fibers [20, 31, 32]; the greater the size of crystallites and ω _c , the greater the tensile modulus. We can find that the MJ series carbon fibers have larger size of crystallites and ω _c than the T series’. Figure 4c shows that the tensile modulus increases linearly with increasing ω _v, and five curves are almost horizontal lines. Elongated microvoids in carbon fibers along the fiber axis result in a high tensile modulus. On the contrary, dumpy microvoids can degrade the properties of carbon fibers. Figure 4d illustrates the tensile modulus plotted against the volume fractions of crystallites f ₁. It is found that the tensile modulus shows a non-linear behavior with f ₁. Usually, the greater the f ₁, the higher the tensile modulus. Figure 4e shows that the tensile modulus increases linearly with descending f ₂. It is obvious that the reduction of microvoids is good for the increase of the tensile modulus of carbon fibers, because the reduced volume of microvoids whose tensile modulus is zero transformed either crystallites or amorphous components or both of them.

Fig. 4

Relationships between the tensile modulus and the factors for the bulk PAN-based carbon fibers

Factors comparisons

The five factors Π, ω _c, ω _v, f ₁, and f ₂, which affect the tensile modulus of PAN-based carbon fibers, are discussed. Figure 5 shows the tensile modulus against the five factors for T300, T700, T800, M35 J, and M40 J (note that the curve of f ₂ is in the second quadrant of the coordinate system, see Fig. 4e, we transform it into the first quadrant for comparison). In Fig. 5, we can see that ω _c, f ₂, and ω _v have a smaller influence on the tensile modulus by considering the slope of curves. We can see that ω _c most affects the tensile modulus and f ₂ is the next and ω _v is the last. Π and f ₁ have a greater effect on tensile modulus than ω _c, f ₂, and ω _v except for Π of M40J within the interval 12 % ≤ Π ≤ 15 %, see Fig. 5e. Compare Π with f ₁, and we can find that Π has a greater effect than f ₁ in initial; with the increase of Π and f ₁, the effect of Π on tensile modulus is surpassed by f ₁ as shown Fig. 5a–e.

Fig. 5

Comparison between the factors for the bulk PAN-based carbon fibers

Conclusions

Our new three-phase micromechanical model which is established by applying the Eshelby equivalent inclusion theory and Mori–Tanaka’s mean stress method can be successfully employed to predict the tensile modulus for carbon fibers. Using this micromechanical model, the volume fractions of crystallites of carbon fibers can be calculated which is able to apply to the quantitative calculation. It is noted that the present study has important implications for understanding the microstructural parameters that control the tensile modulus of PAN-based carbon fibers, such as the orientation degree of crystallites, volume fractions of crystallites and microvoids, and aspect ratios of crystallites and microvoids. By analyzing the new three-phase micromechanical model, we derive the tensile modulus of carbon fibers increase with increasing the orientation degree of crystallites, volume fractions of crystallites, aspect ratios of crystallites and microvoids, and the tensile modulus of carbon fibers increase with decreasing the volume fractions of microvoids. Comparing the five factors that influenced the tensile modulus of carbon fibers, we find that the volume fractions of crystallites and the aspect ratios of crystallites have the most significant effect for tensile modulus. The present study can provide useful help for improving mechanical properties and in the manufacturing process of carbon fibers.

Appendix: coordinate transformation

Consider vector u in the global coordinate system O − x ₁ x ₂ x ₃ and u ^′ in the local coordinate system $O - x^{\prime}_{ 1} x^{\prime}_{ 2} x^{\prime}_{ 3}$. See Fig. 6.

Fig. 6

Diagram of the global coordinate system and local coordinate system

The transformation relation between two coordinate systems can be expressed u ^′ = Qu, where Q is a Transformation tensor. It can be written in matrix form as

$Q = \left[ {\begin{array}{*{20}c} {\cos \theta \cos \varphi } & { - \cos \theta \sin \varphi } & {\sin \theta } \\ {\sin \varphi } & {\cos \varphi } & 0 \\ { - \sin \theta \cos \varphi } & {\sin \theta \sin \varphi } & {\cos \theta } \\ \end{array} } \right].$

For second-order stress tensors σ in the global coordinate system and σ ^′ in the local coordinate system, we have σ = Q ^T σ ^′ Q. Similarly, ɛ = Q ^T ɛ ^′ Q, where ɛ and ɛ ^′ are second-order strain tensors in the global coordinate system and the local coordinate system, respectively.

Stress tensors and strain tensors can also be written in vectors form

$$ \sigma = (\sigma _{1} \sigma _{2} \sigma _{3} \sigma _{4} \sigma _{5} \sigma _{6} )^{T} ,\quad \varepsilon = (\varepsilon _{1} \varepsilon _{2} \varepsilon _{3} \varepsilon _{4} \varepsilon _{5} \varepsilon _{6} )^{T} $$

$$ \sigma^{\prime}\,\left( {\sigma_{1}^{{\prime }} \sigma_{2}^{{\prime }} \sigma_{3}^{{\prime }} \sigma_{4}^{{\prime }} \sigma_{5}^{{\prime }} \sigma_{6}^{{\prime }} } \right)^{\text{T}} \,\varepsilon^{{\prime }} = \left( {\varepsilon_{1}^{{\prime }} \varepsilon_{2}^{{\prime }} \varepsilon_{3}^{{\prime }} \varepsilon_{4}^{{\prime }} \varepsilon_{5}^{{\prime }} \varepsilon_{6}^{{\prime }} } \right)^{\text{T}}; $$

then the transformations of stress vectors and strain vectors are σ = Tσ ^′ and $\varepsilon = \left( {T^{ - 1} } \right)^{\text{T}} \varepsilon^{{\prime }},\;$where T and $\left( {T^{ - 1} } \right)^{\text{T}}$ are the stress transformation matrix and strain transformation matrix, T ⁻¹ is inverse of the T, and $\left( {T^{ - 1} } \right)^{\text{T}}$ is transpose of the T ⁻¹. T and $\left( {T^{ - 1} } \right)^{\text{T}}$ can be expressed as

$$ T = \left[ {\begin{array}{*{20}c} {T_{A} } & {2T_{B} } \\ {T_{C} } & {T_{D} } \\ \end{array} } \right],\,\left( {T^{ - 1} } \right)^{\text{T}} = \left[ {\begin{array}{*{20}c} {T_{A} } & {T_{B} } \\ {2T_{C} } & {T_{D} } \\ \end{array} } \right], $$

where

$$ T_{\text{A}} = \left[ {\begin{array}{*{20}c} {\cos^{2} \theta \cos^{2} \varphi } & {\sin^{2} \varphi } & {\sin^{2} \theta \cos^{2} \varphi } \\ {\cos^{2} \theta \sin^{2} \varphi } & {\cos^{2} \varphi } & {\sin^{2} \theta \sin^{2} \varphi } \\ {\sin^{2} \theta } & 0 & {\cos^{2} \theta } \\ \end{array} } \right], $$

$$ T_{\text{B}} = \left[ {\begin{array}{*{20}c} { - \sin \theta \cos \varphi \sin \varphi } & { - \cos \theta \sin \theta \cos^{2} \varphi } & {\cos \theta \cos \varphi \sin \varphi } \\ {\sin \theta \cos \varphi \sin \varphi } & { - \cos \theta \sin \theta \sin^{2} \varphi } & { - \cos \theta \cos \varphi \sin \varphi } \\ 0 & {\cos \theta \sin \theta } & 0 \\ \end{array} } \right], $$

$$ T_{\text{C}} = \left[ {\begin{array}{*{20}c} { - \cos \theta \sin \theta \sin \varphi } & 0 & {\cos \theta \sin \theta \sin \varphi } \\ {\cos \theta \sin \theta \cos \varphi } & 0 & { - \cos \theta \sin \theta \cos \varphi } \\ { - \cos^{2} \theta \cos \varphi \sin \varphi } & {\cos \varphi \sin \varphi } & { - \sin^{2} \theta \cos \varphi \sin \varphi } \\ \end{array} } \right], $$

$$ T_{\text{D}} = \left[ {\begin{array}{*{20}c} {\cos \theta \cos \varphi } & { - \cos^{2} \theta \sin \varphi - \sin^{2} \theta \sin \varphi } & {\cos \varphi \sin \theta } \\ {\cos \theta \sin \varphi } & {\cos^{2} \theta \cos \varphi - \sin^{2} \theta \cos \varphi } & {\sin \theta \sin \varphi } \\ { - \cos^{2} \varphi \sin \theta + \sin \theta \sin^{2} \varphi } & {2\cos \theta \sin \theta \cos \varphi \sin \varphi } & {\cos \theta \cos^{2} \varphi - \cos \theta \sin^{2} \varphi } \\ \end{array} } \right]. $$