PSI - Issue 2_B

Jaime Planas et al. / Procedia Structural Integrity 2 (2016) 3676–3683 J. Planas et al. / Structural Integrity Procedia 00 (2016) 000–000

3681

6

σ

10 D

C

D

D

RIGID SUBSTRATE

(a)

(b)

z

Fig. 7. Dimensions and boundary conditions of the shrinking slab (a), and elastic shrinkage stress profile (b); the nominal shrinkage stress σ S , the ideal elastic stress at the top surface, is taken as the main loading variable.

σ

1-4 f t

σ

f t

(a)

(b)

D /2

linear shrinkage profile

D

linear softening

w

z

w 1

(c)

Fig. 8. Numerical settings: (a) linear softening curve; (b) linear shrinkage profile, with nominal shrinkage stress σ S increasing from f t to 4 f t ; (c) finite element mesh used in the computations.

4. Numerical study of a slab subjected to thermal or hygral shrinkage

A simplified, two-dimensional analysis of a shhrinking slab is considered next. The geometry of the slab is shown in Fig. 7(a), and the uniaxial stress field that would be generated in the absence of cracking (i.e., in elastic regime), is sketched in Fig. 7(b). The shrinkage stress at the top of the slab defines the strength of the shrinkage, and the shape of the stress profile defines its depth. We can, conventionally, define C as a characteristic shrinkage depth. Obviously σ S / C is the measure of the shrinkage gradient at the top surface of the slab. The problem has been further simplified by using a linear softening function (Fig. 8a) and a constant depth linear shrinkage profile (Fig. 8b). A relatively fine mesh has been used as shown in Fig. 8(c), with 7539 nodes and 15078 elements. The crack patterns were computed when the shrinkage stress σ S increases f t to 4 f t , for a slab with a depth D = 0 . 3 2 . The calculations were carried out in 30, 6 and 3 steps, which correspond to shrinkage strength steps ∆ σ S of 0 . 1 f t , 0 . 5 f t and f t , respectively. Figure 9(a) shows the evolution of the crack pattern for 10 selected steps in the case where the calculation was carried out in 30 steps. In the images, the maximum principal stress is drawn for each element and the crack lines are superimposed. As in all the figures of the paper, dark red represents the tensile strength, and dark blue corresponds to zero stress. In steps 1 and 3, the crack distribution is nearly uniform (within a fraction of the finite element size, obviously). In step 5, some cracks are seen to protrude more than the rest, but the crack length and spacing is still quite uniform. In step 7, triangular unloading zones can be seen coinciding with the root of the longer cracks, which seem to be localized. In step 9, one of every 2 cracks, on average, grow faster and the stress front becomes clearly wavy at the separation between the yellow band and the light-red band, corresponding to a stress of 0 . 6 f t . In step 11 the waviness increases and the shielding of the cracks between the longer cracks becomes apparent. In steps 15 to 30 the unloading triangles at the roots of the main cracks become apparent and evolve from yellow to dark blue, but