21.2.4 Two-phase Model (Walraven)

The two-phase model, proposed by Walraven [110], is based on the following assumptions:

• The concrete is regarded as a two-phase material, with perfectly stiff spherical inclusions and a perfectly plastic matrix.
• The grading of the aggregate matches Fuller's curve.
• The active contact areas between the inclusions and the matrix are related to interface displacements via geometric relations and take into account the statistics of aggregate distribution.
• The compressive contact strength of the matrix is related to the concrete strength while the shear contact strength is related linearly to the compressive contact strength via a constant friction coefficient.

Walraven has developed this theoretical model for pure aggregate interlock, i.e., aggregate interlock in cracks which are not intersected by reinforcing bars. Figure 21.11

Figure 21.11: Two-phase model (Walraven)

shows the response diagram for this model. Shear stress and normal stress are obtained from equilibrium when a given tangential and normal crack displacement occurs. The formulation is given by

$$\sigma_{{\mathrm{pu}}}^{} \mu \sigma_{{\mathrm{pu}}}^{} \mu$$ (21.19)

where A_n and A_t are the averaged contact areas in the directions n and t between the inclusions and the matrix. $\sigma_{{\mathrm{pu}}}^{}$ is the matrix compressive strength and $\mu$ is the coefficient of friction between the inclusion and the matrix. The tangential stiffness terms are functions of the crack displacement  dt , the normal crack displacement $\Delta$u_n and the distribution of the aggregate, see Feenstra [26].