21.2.1 Rough Crack Model (Bazant & Gambarova)

Bazant & Gambarova [6] introduced a rough crack model by considering the crack surface as a regular array of trapezoidal asperities. Figure 21.8

Figure 21.8: Rough crack model (Bazant & Gambarova)

shows the response diagram of this model which has been used merely in qualitative sense, i.e., to introduce the general properties to be expected:

• The wedging effects of the interface asperities make the shear stress primarily dependent on the displacement ratio r =  dt/$\Delta$u_n .
• For large values of the displacement ratio r , the shear stress must exhibit an asymptote because of micro-cracking and crushing in the mortar close to the aggregate particles.
• For large values of the normal crack displacement, the contact at the interface is lost, $\Delta$u_n > ${\tfrac{{1}}{{2}}}$D_max , where D_max is the maximum aggregate size.

The constitutive model is determined by optimizing the fits of Paulay & Loeber's [82] test results at constant crack width. The relations are

$$\tau_{{u}}^{} {\dfrac{{ a_{3} + a_{4} \vert r \vert^{3} }}{{ 1 + a_{4} r^{4} }}} {\dfrac{{ a_{1} }}{{\Delta u_{n}}}} \left(\vphantom{ a_{2} \vert f_{t} \vert }\right. \left.\vphantom{ a_{2} \vert f_{t} \vert }\right)^{{p}}_{}$$ (21.16)

with

$$\left(\vphantom{ 1 - \frac{ 0.231 }{ 1 + 0.185 \Delta u_{n}+ 5.63 ( \Delta u_{n})^{2} } }\right. {\frac{{ 0.231 }}{{ 1 + 0.185 \Delta u_{n}+ 5.63 ( \Delta u_{n})^{2} }}} \left.\vphantom{ 1 - \frac{ 0.231 }{ 1 + 0.185 \Delta u_{n}+ 5.63 ( \Delta u_{n})^{2} } }\right)$$ p

$${\frac{{ \,\mathrm{d}t}}{{ \Delta u_{n}}}}$$ r

$$\tau_{{u}}^{} {\frac{{ \tau_{0} a_{0} }}{{ a_{0} + ( {\Delta u_{n}} )^{2} }}}$$

a₀ = 0.01D_max²        a₁ = 0.000534        a₂ = 145.0        

$${\frac{{ 2.45 }}{{ \tau_{0} }}} \left(\vphantom{ 1 - \frac{ 4 }{ \tau_{0} } }\right. {\frac{{ 4 }}{{ \tau_{0} }}} \left.\vphantom{ 1 - \frac{ 4 }{ \tau_{0} } }\right)$$ a₃

$$\tau_{{0}}^{}$$ = 0.245f_c = 0.195f_cc

The notation f_c is used for the compressive cylindrical strength of the concrete, and the more frequently used compressive cube strength is denoted by f_cc . For the tangential stiffness coefficients see Feenstra [26].