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. $y=mx,\text{\hspace{0.17em}}m={y}_{a}/{x}_{a}$ $\left(0\le x\le {x}_{a}\right)$ 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. $y=mx+c,\text{\hspace{0.17em}}m=\frac{1-{y}_{a}}{{x}_{b}-{x}_{a}},\text{\hspace{0.17em}}c=1-m{x}_{b}$ $\left({x}_{a}\le x\le {x}_{b}\right)$ 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. $y=mx+c,\text{\hspace{0.17em}}m=\frac{{y}_{c}-1}{{x}_{c}-{x}_{b}}\frac{{y}_{2}-1}{{x}_{2}-1},\text{\hspace{0.17em}}c=1-{x}_{b}$ $\left({x}_{b}\le x\le {x}_{c}\right)$ 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. $y=mx+c,\text{\hspace{0.17em}}m=\frac{{y}_{d}-{y}_{e}}{{x}_{d}-{x}_{e}},\text{\hspace{0.17em}}c={y}_{d}-m{x}_{d}$ $\left({x}_{d}\le x\le {x}_{e}\right)$ 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. $y=mx+c,\text{\hspace{0.17em}}m=\frac{{y}_{f}-{y}_{g}}{{x}_{f}-{x}_{g}},\text{\hspace{0.17em}}c={y}_{f}-m{x}_{f}$ $\left({x}_{f}\le x\le {x}_{g}\right)$ 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). $y={y}_{g}\left[1-{e}^{\left({x}_{g}-1\right)}\right]$ $\left({x}_{g}\le x\le {x}_{h}\right)$