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

3678

3

The present work concentrates on two simple “clean” cases to find out whether the computational procedure de veloped by Sancho et al. (2007a,b) and used in Sanz et al. (2013) might hide spurious sensitivity to mesh or step size. The method is outlined in Sec. 2, and the results for a three-point bend beam and for a shrinking slab are presented and discussed in Sec. 3 and 4, respectively; a few conclusions are drawn in Section 5, which closes the paper.

2. Background

Figure 2 displays the main ingredients of the numerical and conceptual model used in the computations (Sancho et al., 2007a). The finite elements are constant strain triangles with an embedded cohesive crack. Strong discontinuity kinematics is imposed, which means, as shown in the figure, that the crack cuts only two sides of the element and, therefore, the inelastic strain of the side AB is zero: this is the main di ff erence with respect to the traditional smeared crack approach. For the detailed procedures for crack initiation and growth, the reader is referred to the original paper by Sancho et al. (2007a). Other essential features of the model are as follows: (1) the crack is required to satisfy local equilibrium, which implies that its exact position in the element is not required; (2) the cohesive traction vector t and the crack opening vector w are collinear: this was initially just a convenient assumption, but nowadays is known to be a fundamental requirement for the cohesive model to be frame-indi ff erent (Costanzo, 1998); (3) as a consequence, the crack behavior is completely defined by the softening curve for pure crack opening, as determined in classical fracture tests; (4) a damage-like unloading to the origin is the last ingredient required to be able to write the full traction-separation law —see Sancho et al. (2007a) for details. The computations are strictly local, i.e., no crack tracking algorithm is used, the orientation of the crack in the element and the corresponding solitary node are determined based only of the current nodal displacements in the corresponding element. However, two numerical-only features are used to avoid crack locking: (1) The initial elastic stti ff nes matrix is used for the element throughout the computation, and (2) the crack in the element is allowed to reorient itself following the principal stress rotations while its crack opening is small compared to w 1 (this is called limited crack adaptability). Feature (1) implies a large number of iterations (but these are very fast because back substitution is only required), which turns out to be a virtue when combined with feature (2) since the cracks are given more opportunities to adopt the right orientation (or even get closed), while seeking for convergence, in a kind of self-annealing process.

traction

|

t |

gradient of shape function of ‘solitary’ node

f t

n

w

t

| t | = f ( | w | )

b +

B

A

w 1

crack opening

˜ w

| w |

Fig. 2. Embedded crack model.

3. Numerical study of a three-point bent beam

The geometry and material data of the beam are identical to those described in Sec. 4.1 of Planas et al. (2003). Figure 3 defines the geometry of the beam and of the softening curve. All the analyses are carried out in dimensionless form, and the only relevant parameter is the relation D / 2 where 2 is the second characteristic length defined as

Ew 1 f t

(1)

2 =

.

Note: the second characteristic length is similar to Hillerborg’s characteristic length ch : = EG F / f 2 initial part of the softening curve only; ch = 2 when G F = f t w 1 , which is close to reality for concrete.

t , but uses properties of the