21.2.5 Contact Density Model (Li et al.)

The Contact Density model is based on two proposals and three assumptions by Li et al. [59] which can be summarized as follows.

• A crack plane consists of a number of areas (contact units) with various inclinations. These inclinations $\theta$ from - ${\tfrac{{1}}{{2}}}$$\pi$ to ${\tfrac{{1}}{{2}}}$$\pi$ can be described by a contact density probability function $\Omega$($\theta$) .
• The direction of each contact stress is proposed to be fixed and normal to the initial contact direction denoted as $\theta$ .
• The density function $\Omega$($\theta$) is assumed as a trigonometric function which is independent of the size and the grading of the aggregate, and of the strength and kinds of coarse aggregates.
• The contact force is computed with a simple elasto-perfectly plastic model for the contact stress prediction $\sigma_{{\mathrm{con}}}^{}$ .
• The effective ratio of contact area K($\Delta$u_n) expresses the loss of contact when the normal crack displacement $\Delta$u_n is large enough compared with the roughness of the crack surface.

Figure 21.12: Contact density model (Li et al.)

Figure 21.12 shows the response diagram for this model. The mathematical formulation is given by

$$\begin{split}f_{t} &= \int_{ - \tfrac{1}{2}\pi }^{ \tfrac{1}{... ...t} \Omega ( \theta ) \cos \theta \,\mathrm{d}\theta \end{split}$$ (21.20)

in which the surface area of the crack A_t is 1.27 x the sectional area of the crack plane. For derivation of the stiffness coefficients of this model see Feenstra [26].