21.1.2 Linear Tension Softening

In case of linear tension softening [Fig.21.4],

Figure 21.4: Linear tension softening

the relation of the crack stress is given by

$${\frac{{f_{n} (\Delta u_{n})}}{{f_{\mathrm{t}}}}} \begin{cases} 1 - \dfrac{\Delta u_{n}}{\Delta u_{n.\mathrm{u... ... $\Delta u_{n.\mathrm{ult}}< \Delta u_{n}< \infty$} \end{cases}$$ (21.9)

with the ultimate crack strain

$$\Delta {\frac{{ G_{\mathrm{f}}^{\mathrm{I}} }}{{ f_{\mathrm{t}} }}}$$ (21.10)

Unloading and reloading

can be modeled according to a secant approach or an elastic approach. In the secant approach, the relation between the traction and the relative normal displacement is linear up to the origin, after which the initial stiffness is recovered. In the elastic approach, the initial stiffness is recovered immediately after the relative normal displacement has become less than the current maximum relative normal displacement [Fig.21.4].