An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

Abstract

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...

An experimental and numerical study of the pattern of cracking of concrete due to steel reinforcement corrosion

“…More generally, we take the threshold to be Wth = a'wi (1) and call a' the adaptation factor of the embedded crack. Further details can be found in [6,7].…”

“…Concrete cracking is numerically modeled using a relatively simple implementation which combines constant strain elements with an embedded cohesive crack with limited adaptability [6,7]. The cohesive crack used in the element formulation is the simplest 3D extension of the standard cohesive crack under pure opening mode.…”

“…In the present research, the calculations were driven by the corrosion depth, which means that the model does not depend on time explicitly, but it depends on it implicitly since the corrosion depth does depend on Poisson's ratios of the oxide and the steel 0"/ maximum principal stress T shear stress applied to the oxide surface time through the corrosion current density. The expansive joint element is implemented within the finite element framework COFE (Continuum Oriented Finite Elements) and it works together with finite elements with an embedded adaptable cohesive crack [6,7] which follow the basic cohesive model proposed by Hillerborg et al [8] to simulate the fracture of concrete and has been since used by many other researchers; see,e.g., [9][10][11] for details of the model and [12,13] for recent developments in the testing procedures to obtain the bilinear softening curve.…”

An embedded cohesive crack model for finite element analysis of quasi-brittle materials

“…As in previous works [41][42][43][44], based on this approach, the deformation is localised on a line using the concept of the cohesive crack, and the discrete constitutive relation for mixed mode fracture is a cohesive crack with a central-force model. The model avoids crack tracking [45,60] or exclusion zones for crack growing.…”

“…For this purpose the crack is numerically implemented as a discontinuity embedded in a classical FE. The modelling is mainly based on the authors' proposal for concrete [41][42][43] and masonry [44].…”