21.1.3 Nonlinear Tension Softening (Hordijk et al.)

Hordijk, Cornelissen & Reinhardt [17,43] proposed an expression for the softening behavior of concrete which also results in a crack stress equal to zero at a crack width $\Delta$u_n.ult [Fig.21.5].

Figure 21.5: Nonlinear tension softening (Hordijk et al.)

The function is defined by

$${\frac{{f_{n} (\Delta u_{n})}}{{f_{\mathrm{t}}}}} \begin{cases} \left( 1 + \left( c_{1} \dfrac{\Delta u_{n}}{\... ... $\Delta u_{n.\mathrm{ult}}< \Delta u_{n}< \infty$} \end{cases}$$ (21.11)

with the parameters c₁ = 3 and c₂ = 6.93 and with the ultimate crack strain defined by

$$\Delta {\frac{{ G_{\mathrm{f}}^{\mathrm{I}} }}{{ f_{\mathrm{t}} }}}$$ (21.12)

Unloading and reloading

can be modeled according to a secant approach, an elastic approach or by application of hysteresis. 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.21.5]. The third possibility is to apply the hysteresis model of Hordijk [43,49] in which unloading and reloading follow different paths [Fig.21.6].

Figure 21.6: Hysteresis model (Hordijk)