21.2 Crack Dilatancy

The crack can be considered as open when its normal relative displacement $\Delta$u_n has become greater than the ultimate magnitude of the normal relative displacement $\Delta$u_n.ult of a softening model. For such an open crack, the constitutive model of a rough crack can be utilized. The constitutive relation of the rough, open crack is mobilized when the displacement tangential to the crack faces has become greater than zero, in the absolute sense. Consider an open crack which is planar but microscopically rough [Fig.21.7].

Figure 21.7: Rough crack

The global crack displacements $\Delta$u_n and  dt are the relative displacements of the two parts of the structure, separated by the crack. With this definition, the global crack width is independent of the global crack sliding, but the local sliding and width will vary along the crack, depending on the crack geometry.

Due to the complexity of the problem, the constitutive laws for crack dilatancy which have been proposed by various authors, are mostly based on a total deformation theory. This theory expresses the tractions as a function of the total relative displacements, see for instance Bazant & Gambarova [6],

$$\begin{cases} t_{n} &= f_{n} (\Delta u_{n},\,\mathrm{d}t) \\ [2ex] t_{t} &= f_{t} (\Delta u_{n},\,\mathrm{d}t) \end{cases}$$ (21.14)

Differentiating (21.14) results in expressions for the crack stiffness coefficients:

$$\begin{cases} D_{11} &= \dfrac{ \partial f_{n} } { \partial\... ...= \dfrac{ \partial f_{t} } { \partial\,\mathrm{d}t} \end{cases}$$ (21.15)

The mathematical models for crack dilatancy can be classified into two categories. The first category is based on experimental results and has an empirical formulation, we will denote it as empirical crack models. The second category is based on an assumption of the shape of the crack surface and has a rational formulation, we will denote this category as physical crack models. Although there are many models which give good results, the ones supported by DIANA have been restricted to a few models that are characteristic of their class.

Empirical crack models.

In this category DIANA supports a Rough crack model according to Bazant & Gambarova [§21.2.1], another Rough crack model according to Gambarova & Karakoç [§21.2.2], and an Aggregate interlock relation according to Walraven & Reinhardt [§21.2.3].

Physical crack models.

In this category DIANA supports a Two-phase model proposed by Walraven [§21.2.4] and Contact density model proposed by Li et al. [§21.2.5].