### Calculation of impregnation ratio of elliptical model (Kobayashi et al. 2017)

In order to fully demonstrate the mechanical properties of reinforcing carbon fibers in CFRTP, complete impregnation of the resin to reinforcing fiber yarns is necessary. Thus it is an important to analytically-predict the time necessary for complete resin impregnation to yarns. In the present study, resin impregnation behavior was analytically predicted similar to the previous study (Kobayashi et al. 2017).

The fiber yarn is considered as a porous medium, where the gap between fibers is regarded as pore. In general, the impregnation phenomenon of the rein to fiber yarns is regarded as laminar flow to a porous medium and represented using the Darcy’s law (Åström et al. 1992). The Darcy’s law is expressed as,

$$ u=-\frac{k}{\mu}\cdot \frac{\partial P}{\partial x} $$

(1)

where *u* is Darcy’s velocity, *μ* is the viscosity, *∂P / ∂x* is the pressure gradient, and *k* is the permeability coefficient.

Also, an equation of continuity can be written as,

$$ \nabla \cdot \mathbf{u}=0 $$

(2)

In the present study, the cross-section of a fiber yarn was deformed to an elliptical shape by compression loading. Thus it is assumed that the fiber yarn has a cross section close to an elliptical shape with major radius *a*_{0} (*x* direction) and minor radius *b*_{0} (*y* direction), as shown in Fig. 2. The fiber longitudinal direction is defined as *z*.

As shown in Fig. 2, an oval un-impregnated region where major and minor radii are *a*_{1} and *b*_{1} respectively, is assumed. *a*_{1} and *b*_{1} become shorter with molding time and resultant resin impregnation.

By using the fiber volume fraction *V*_{f} in the fiber yarn and the eq. (1), Darcy’s velocity is converted to flow front velocity in *x* direction.

$$ \left(1-{V}_f\right)\frac{da_1}{dt}={\left.{u}_x\right|}_{x={a}_1}=-\frac{k}{\mu}\cdot {\left.\frac{\partial P}{\partial x}\right|}_{x={a}_1} $$

(3)

By re-arranging and integrating both sides,

$$ {a}_1=-\frac{k}{\mu \left(1-{V}_f\right)}\int \frac{\partial P}{\partial x} dt $$

(4)

In the same way, the following equation is obtained for the *y* direction.

$$ {b}_1=-\frac{k}{\mu \left(1-{V}_f\right)}\int \frac{\partial P}{\partial y} dt $$

(5)

In the elliptical model as shown in Fig. 2, the cross-sectional area of the fiber yarn and the cross-sectional area of the non-impregnated region are *S*_{0} = π*a*_{0}*b*_{0} and *S*_{1} = π*a*_{1}*b*_{1}, respectively, so that the impregnation ratio *I* becomes.

$$ I(t)=1-\frac{\pi {a}_1{b}_1}{\pi {a}_0{b}_0}=1-\frac{a_1(t){b}_1(t)}{a_0{b}_0} $$

(6)

The pressure gradient of the coordinates (*a*_{1}, 0) and (0, *b*_{1}) at the flow front of the resin shown in Fig. 2 should be obtained to calculate the position of the flow front from eqs. (4) and (5) and resultant impregnation ratio represented by eq. (6) at a certain time *t*.

### Calculation of pressure gradient in elliptical model

In the present study, resin flow in the axial direction is neglected. In orthogonal coordinates as shown in Fig. 2, since the flow velocity *u*_{z} in the *z* direction is assumed as 0, eq. (2) can be expressed as follows.

$$ \nabla \mathbf{u}=\frac{\partial {u}_z}{\partial x}+\frac{\partial {u}_y}{\partial y}=0 $$

(7)

Here, substituting eq. (3) into eq. (7) and assuming that the melt viscosity *μ* and the permeability coefficient *k* of the resin are independent of time, the following equation is obtained.

$$ \frac{\partial^2P}{\partial {x}^2}+\frac{\partial^2P}{\partial {y}^2}=0 $$

(8)

In this case, the boundary condition for the elliptical model as shown in Fig. 2 is defined as follows.

1: *P*=*P*_{m} (resin pressure) on the outer boundary of the elliptical fiber yarn \( \left(\frac{x^2}{a_0}+\frac{y^2}{b_0}=1\right) \).

2: *P*=*P*_{0} (atmospheric pressure) on the boundary of the elliptical un-impregnated region \( \left(\frac{x^2}{a_1}+\frac{y^2}{b_1}=1\right) \).

Since the elliptical model shown in Fig. 2 is symmetrical to the *x* and *y* axes, the pressure distribution for the resin impregnation to the fiber yarn in the 1/4 model as shown in Fig. 3 shall be considered. The pressure gradient at the flow front could be obtained from the pressure distribution. It is, however, difficult to solve eq. (8) analytically, a mathematical approach was carried out by using the boundary element method.

Eqs. (4) and (5) are discretized in terms of time *t*, when *t* = *t*_{i} = *iΔT* (*i*: natural number), as

$$ {a}_1\left({t}_i\right)={a}_0-{\sum}_{j=1}^{i-1}{\left.\frac{k}{\mu \left(1-{V}_f\right)}\frac{\partial P}{\partial x}\right|}_{x={a}_1\left({t}_j\right)}\bullet \Delta t $$

(9)

$$ {b}_1\left({t}_i\right)={b}_0-{\sum}_{j=1}^{i-1}{\left.\frac{k}{\mu \left(1-{V}_f\right)}\frac{\partial P}{\partial y}\right|}_{x={b}_1\left({t}_j\right)}\bullet \Delta t $$

(10)

where *a*_{0} and *b*_{0} are the positions of flow front at *t* = 0 considering capillary effect.

### Permeability

In the present study, permeability, *k*, is calculated according to the Kozeny-Carman equation (Gutowski 1985) as,

$$ k=\frac{{d_f}^2}{16{k}_0}\frac{{\left(1-{V}_f\right)}^3}{{V_f}^2} $$

(11)

where *d*_{f} is the diameter of a single fiber and *k*_{0} is the Kozeny constant. This equation is known as the permeable Kozeny-Carman equation.

Generally, the fiber volume fraction is related to the applied pressure. Gutowski (1985) derived this relationship in consideration of the elastic deformation of the fiber yarn. Assuming a quasi-static loading, the relationship between pressure and fiber volume fraction is expressed as,

$$ {P}_m-{P}_0=A\frac{\sqrt{\frac{V_f}{V_0}}-1}{{\left(\sqrt{\frac{V_a}{V_0}}-1\right)}^4} $$

(12)

where *P*_{a} is the molding pressure, *P*_{0} is the atmospheric pressure, *A* is the experimental spring constant, *v*_{a} is the maximum possible fiber volume fraction, and *v*_{o} is the no-load volume fraction.

### Correction of resin infiltration ratio by capillary phenomenon

The impregnation ratio can be obtained with respect to time based on the theory described above. On the other hand, the actual resin impregnation to a fiber yarn occurs before temperature and pressure reach the target values because of capillary action. For example, resin impregnation was confirmed at molding time = 0 s as described below. In this study, the influence of capillary action is not analytically considered. In order to consider the capillary action semi-quantitatively, the predicted curve was sifted in the time direction as shown in Fig. 4. Here the time shift is defined as *M* value. Effect of molding condition on the *M* values are also discussed later.