ABSTRACT
In quasi-brittle materials, the discontinuities are typically much softer than the bulk and its opening can be considered to be transmitted to the neighbouring material as if it were a rigid body movement. The stresses along the discontinuity are in equilibrium with the surrounding bulk, which unloads elastically as the discontinuity widens and softens its tensile stress. Under this assumption, it can be shown that the discretised set of equations for a finite element
1 INTRODUCTION
Estimating the real stiffness of a partially fractured concrete member for a given design load is a very difficult task, in particular because the concrete placed between cracks has a significant role in transferring tensile stresses-this is usually known as the ‘tension stiffening effect’—and the crack pattern can be random and it also depends on the maximum load the structure experienced. For this reason, predicting the behaviour of a concrete member subjected to serviceability loads, and corresponding crack openings and overall deformation, can encompass several difficulties. In the last decades, the research community has witness many computational approaches to be introduced for predicting the fractured behaviour of different materials with the discrete representation of cracks. There are now different numerical techniques that can be used. Advanced techniques, for instance, are able to avoid the need for remeshing when cracks propagate by making use of enriched finite element meshes (nodes or elements) that can simulate the discontinuous displacement fields associated with cracks (Belytschko & Black, 1999, Moës et al., 1999, Wells & Sluys, 2001a, Areias & Belytschko, 2005, Dvorkin et al., 1990, Oliver, 1996, Jirásek & Zimmermann, 2001, Wells & Sluys, 2001b, Linder &
containing a crack is given by (Simo & Rifai, 1990, Dias-da-Costa et al., 2009, Dias-da-Costa et al., 2013, Dias-da-Costa et al., 2010):
d
⎡ ⎤ ⎧ ⎫ ⎧ ⎫⎪ed ⎪ ⎪d eˆ⎢ ⎥ ⎨ ⎬ = ⎨ ⎬⎢ ⎥ ⎪ ⎪d e ⎪ ⎪e⎩ ⎭⎣ ⎦d p ⎩ ⎭ ⎪
−
(1)
lar finite element, Be is the strain-nodal displacement matrix, De is the linearised constitutive matrix for the bulk, K N T Nd weT we
stiffness of the discontinuity, Nwe contains linear interpolation functions defined along the discontinuity, Te is the linearised constitutive matrix for the discontinuity, K pe is a penalty matrix that enforces the shear jump transmission along the discontinuity,
placements on the regular nodes of the element, we is a vector containing the opening of the discontinuity at both extremities over the edges of the element and ˆ ef are the regular forces. It can be shown that ( ) ˆd d dd Te e e ek ew w f is zero under certain circumstances.
