Stage

Description

Equation

o-a

When a pull-out load $y\le y_a$ , the fibre remain perfectly bonded to the matrix and the interface is kept in an elastic bond condition.

$y=mx,m=y_a/x_a$ $\left(0\le x\le x_a\right)$

a-b

When $y>y_a$ , a part of the fibre is still fully bonded to the matrix while the remaining part is de-bonded.

$y=mx+c,m=\frac{1-y_a}{x_b-x_a},c=1-m{x}_b$

$\left(x_a\le x\le x_b\right)$

b-c

Once interface de-bonding is completed at (b), the horizontal portion of the fibre must overcome kinetic friction as the hooked end of the fibre undergoes reverse bending.

$y=mx+c,m=\frac{y_c-1}{x_c-x_b}\frac{y_2-1}{x_2-1},c=1-x_b$

$\left(x_b\le x\le x_c\right)$

c-d

Due to the cold work from both plastic hinges (PH1 and PH2), the pull-out load remains until the fibre pulled by an additional distance $\left(x_d-x_c\right)$ .

$y=y_c=y_d$ $\left(x_c\le x\le x_d\right)$

d-e

When PH1 has straightened, the fibre moves into the straight part of the channel. Then moving and straightening of PH2 results in a slight decrease in pull-out load.

$y=mx+c,m=\frac{y_d-y_e}{x_d-x_e},c=y_d-m{x}_d$

$\left(x_d\le x\le x_e\right)$

e-f

At this stage, the load remains until the fibre is pulled by an additional distance $\left(x_f-x_e\right)$ .

$y=y_e=y_f$ $\left(x_e\le x\le x_f\right)$

f-g

Once both PH1 and PH2 are completely deformed and straightened, the load drops to (g) with no plastic hinges active.

$y=mx+c,m=\frac{y_f-y_g}{x_f-x_g},c=y_f-m{x}_f$

$\left(x_f\le x\le x_g\right)$

g-h

At this stage, the pull-out load need only overcome kinetic frictional resistance as for a straight fibre. This phase prevails until the whole fibre is completely removed from the matrix at (h).

$y=y_g\left[1-e^{\left(x_g-1\right)}\right]$ $\left(x_g\le x\le x_h\right)$