A B S T R A C T An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi-brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well-known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.Keywords cohesive crack; concrete fracture; embedded crack; finite element method; mixed mode fracture; numerical analysis.
I N T R O D U C T I O NConsiderable effort has been devoted to developing numerical models to simulate the mixed mode fracture of quasi-brittle materials. Traditionally, numerical methods based on the Finite Element Method were classified into two groups 1 : the smeared crack approach and the discrete crack approach, although some authors include a third group: the lattice approach. 2 In the smeared crack approach the fracture is represented in a smeared manner: an infinite number of parallel cracks of infinitely small opening are (theoretically) distributed (smeared) over the finite element. 3 The cracks are usually modelled on a fixed finite element mesh. Their propagation is simulated by the reduction of the stiffness and strength of the material. The constitutive laws, defined by stress-strain relations, are nonlinear and show a strain softening. This approach was pioneered with fixed-crack orthotropic secant models 4-6 and rotating crack models 7-9 .However, strain softening introduces some difficulties in the analysis. The system of equations may become ill-posed, 12-14 and localization instabilities and spurious mesh sensitivity of the finite element calculations may appear. 3 These difficulties can be tackled by supplementing the material model with some mathematical condition. [15][16][17] Other strategies are the non-local continuum models, 18,19 the gradient models, 20 and the micropolar continuum. 21 These procedures are suited to specific problems, but none gives a general solution of the problem. The discrete approach is preferred when there is one crack, or a finite number of cracks, in the structure. The cohesive crack model, developed by Hillerborg et al. 22 for mode I fracture of concrete, was shown to be efficient in modelling the fracture process of quasi-brittle materials. It has been extended to mixed mode fracture (modes I and II) and incorporated into finite element codes 23-28 and into boundary element codes. 29 One...
This paper presents a numerical procedure for fracture of brickwork masonry based on the strong discontinuity approach. The model is an extension of the cohesive model prepared by the authors for concrete, and takes into account the anisotropy of the material. A simple central-force model is used for the stress versus crack opening curve. The additional degrees of freedom defining the crack opening are determined at the crack level, thus avoiding the need of performing a static condensation at the element level. The need for a tracking algorithm is avoided by using a consistent procedure for the selection of the separated nodes. Such a model is then implemented into a commercial code by means of a user subroutine, consequently being contrasted with experimental results. Fracture properties of masonry are independently measured for two directions on the composed masonry, and then input in the numerical model. This numerical procedure accurately predicts the experimental mixed-mode fracture records for different orientations of the brick layers on masonry panels.
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