## Numerical Simulation

The numerical model of the double flat-jack test was built exploiting the symmetry of the problem.

Quadratic elements were used to represent both the brick units and the mortar joints. The failure of both components was assumed as ideal plasticity in compression and linear softening in tension. A fixed smeared crack model based on total deformation was used. All the analyses were performed with the Finite Element Software DIANA 9.1 (de Witte & Schreppers,

2005). The mechanical properties of the materials are summarized in Table 2.

Figure 11a shows the mesh used to model the smallest specimen. Taking advantage of the problem symmetry, only one quarter of the geometry has been discretized. Figure 11b shows details about the loading and boundary conditions. The following procedure has been applied. First, a displacement is imposed to the top of the specimen.

The amount of such displacement can be calculated from another model of the masonry wall ("uncut"), Figure 11. Finite element mesh adopted for Volume 1 exploiting symmetry (crf. Shaded area in Figure 7a). Mesh and materials (a); loads and boundary conditions (b).

without the cut where the flat-jack is placed afterward. This corresponds to the in situ configuration before the test. The imposed displacement is determined such then the vertical stress equals the in-situ value.

Afterwards, the pressure load in both sides of the cut is applied incrementally. When the pressure reaches the in-situ value of the vertical stress, the deformation of the model approaches the configuration obtained from the "uncut" model, exactly like in the experimental procedure.

If the load is increased further, the material comprised in between the two flat-jacks starts to damage. This behavior is caught correctly by the numerical model. Figure 12 shows the stress-strain diagrams obtained for the three different sizes. The arrows indicates the moment at which the horizontal strain suddenly increases, that corresponds to the first vertical cracking.

The compressive strength decreases with increasing the specimen size in a rather good agreement with the experimental tests. On the other hand, the stressstrain path in compression looks a bit stiffer than the experimental one, especially after the cracking occurs.

The crack pattern for the three sizes is shown in Figure 13. It slightly changes varying the size, probably due to the different aspect ratio.

In a previous work (Carpinteri & Lacidogna 2006), a statistical and fractal analysis of data from laboratory experiments was performed, considering the multiscale aspect of cracking phenomena. The fractal criterion takes into account the multiscale character of energy dissipation and the strong size effects associated with it. This makes it possible to introduce a Figure 12. Stress-strain diagrams: Volume 1 (a); Volume 2 (b) and Volume 3 (c). The arrow indicates when first cracking spreads into the specimen.

useful energy-related parameter for the determination of structural damage (as used by Carpinteri et al. 2003, 2004, for reinforced concrete structures) by comparing the AE monitoring results with the values obtained on masonry elements of different sizes tested up to failure by means of double jacks.

Fragmentation theories have shown that, during microcrack propagation, energy dissipation occurs in a fractal domain comprised between a surface and the specimen volume V (Carpinteri & Pugno 2002a, b, 2003). Figure 13. Crack patterns due to flat-jack pressure in the three specimens.

Figure 13. Crack patterns due to flat-jack pressure in the three specimens.

This implies that a fractal energy density (having anomalous physical dimensions):

can be considered as the size-independent parameter. In the fractal criterion of Eq. (4), Wmax = total dissipated energy; r = fractal energy density; and D = fractal exponent, comprised between 2 and 3. On the other hand, during microcrack propagation, acoustic emission events can be clearly detected. Since the energy dissipated, W, is proportional to the number of AE events, N, the critical density of acoustic emission events, Tae, can be considered as a size-independent parameter:

» Cracked elements

Figure 14. Volume effect on Nmax and on the number of cracked finite elements.

Figure 14. Volume effect on Nmax and on the number of cracked finite elements.  