State of the Art

Analysis of fracture processes

Nowadays, the simulation of fracture processes in concrete structures may be performed using two different strategies [1-5]: a continuum approach, where fracture is treated as the end of a process of accumulation and localization of damage without creating a real discontinuity in the material [6-8] and a discrete approach where a displacement discontinuity with an interface with a cohesive constitutive law is introduced at a loading level and with a position, direction and length determined by the application of some physically based criteria [9-13]. Recently, a combined strategy which explores the advantages of both the continuum and the discrete approaches has been introduced to simulate the damage development and the discrete crack propagation in concrete structures [14-16]. In the first stage of the analysis, a continuum damage model is used to describe the evolution of the mechanical properties of the continuum as micro-cracking develops. Then, when the damage variable reaches a critical value [17], a cohesive crack is introduced in the direction of the localization band. The cohesive law to be used in this approach must be calibrated using energetic considerations [18-19]. (top)

Continuum damage mechanics

Damage models can capture the stiffness and strength degradation and the unilateral effects in concrete structures [20-26]. The concept of damage was first introduced by Kachanov [27-28]. Then, Rabotnov [29] defined the effective stresses and introduced the idea of affecting the initial stiffness of the material by a factor depending directly on the actual value of the damage variable. Some years later, thermodynamic laws have been used to set up the theoretical basis for the development of continuum damage mechanics models [30-32].

The isotropic damage models assume a uniform degradation of properties in all directions. In these models only one [33-34] or two [35-36] variables are used to describe the deterioration of the material. For most of the structures and loading conditions their use leads to accurate results. The numerical implementations are usually based on the use of such models, because they combine simplicity and robustness. To take into account the anisotropy induced by the development of diffuse micro-cracking, more sophisticated models have to be considered [38-41]. Damage can also be represented as a second order [42] or fourth order [43] tensor. However, the number of material parameters to be defined and calibrated makes very difficult the use of these more sophisticated models. Damage can be coupled to plasticity models [44-49].

It is well known that standard local constitutive models are inappropriate for materials with strain softening behaviour [50-51]. When, due to the development of damage the material state is in the softening part of the stress-strain curve, the governing differential equations lose ellipticity and the boundary value problem describing the structural response becomes ill-posed. The strain distribution obtained with finite element computations localizes into a narrow band whose width depends on the element size and tends to zero as the mesh is refined. Since damage localizes in a region of zero volume the total amount of energy dissipated vanishes, which is not an admissible situation from the physical point of view. Adequate regularization techniques have to be used to insure the objectivity of the numerical modelling and to recover the well posedness of boundary value problem [52-53].

The fracture energy based regularization is the simplest technique. It takes into account the quantity of energy which must be released in order to propagate a crack and can be implemented by adjusting the post-peak slope of the stress-strain diagram as a function of the element size. When this is done, the energy dissipated in a band of cracking elements does not depend on its width [54-55]. However, this technique presents some limitations: the global behaviour is regularized but the material behaviour remains dependent on the adopted discretization. In general, this type of regularization is used only in the analysis of large structures [56].

A computationally efficient and theoretically sound regularization technique is provided by the concept of non-local averaging [57-62]. In the integral non-local models [63-68] some of the local physical quantities present in the constitutive relation and in the potential of dissipation definition are replaced by corresponding weighted averages defined over the whole domain. The normalized Gauss function [65-67] is usually used as weight function and the procedure must be able to correctly reproduce local uniform fields. As discussed in [2], a correct choice of the physical quantities to be averaged is an important issue to ensure the efficiency and the coherence of the regularization procedure.

Other technique ensuring objectivity is based on the definition of gradient models, where the constitutive relation is enhanced through the definition of gradients of the state variables [69-71]. The gradient models can be classified into two different groups: the explicit [72-74] and the implicit [75-78] models. The explicit models are based on the plasticity gradient models [79-80] and introduce higher order terms in the definition of the potential of dissipation. The implicit models result from mathematical manipulations of the non-local variables. They can be viewed as integral non-local models where the Green function is used in the computation of the averaged quantities [81].

A detailed review of damage models and their evolution can be found in [82-89].

In recent years, non-conventional hybrid and mixed finite element formulations [90-91] for the physically non-linear analysis of concrete structures based on the consideration of isotropic non-local damage models have been developed [92]. In [93-95], the hybrid-mixed stress model based on the use of use of orthonormal Legendre polynomials [96] is used. The stress and the displacement fields in the domain of each element and the displacements on the static boundary are independently approximated. None of the fundamental relations is enforced a priori [97-98] and all field equations are imposed in a weighted residual form, ensuring that the discrete numerical model embodies all the relevant properties of the continuum it represents. The Mazars' isotropic model [33] is used and a non-local integral formulation where the damage variable is taken as the non-local variable is adopted. In [99-101] an improved hybrid-mixed stress model is presented and discussed. The approximation of the stress field in the domain is here replaced by the approximation of the effective stress field. The isotropic model presented by Comi and Perego [64] is now adopted and the strain energy release rate is defined as the non-local variable. An alternative technique based on the definition of an explicit enhanced gradient model has also been tested [92].

The use of hybrid-Trefftz displacement formulations, where the displacements in the domain of each element and the stress field on the kinematic boundary are independently approximated, is reported in [102-103]. The main feature of these models is that the functions used to approximate the displacements are derived from bi-harmonic displacement potentials that solve the Navier equations for a homogeneous elastic material [104-105]. Due to the strategy used in their definition, only while concrete presents a linear elastic behaviour the model may be considered as a pure hybrid-Trefftz formulation. When this behaviour is no longer valid, it becomes an hybrid-displacement model. The use of hybrid-displacement models based on the use of Legendre polynomials is reported in [106]. The isotropic model presented in [64] and the non-local integral formulation based on the computation of strain energy release rate averages is again adopted. (top)

Cohesive fracture mechanics

Linear elastic fracture mechanics is only applicable when the size of the fracture process zone at the crack tip is small when compared to the size of the crack and to the size of the specimen [107]. The cohesive crack model is the simplest technique that overcomes this limitation. In this case, the propagation is governed by a traction- displacement relation across the crack faces near the tip. Hillerborg [108] introduced the concept of fracture energy into the cohesive crack model and proposed a number of traction-displacement relationships for concrete.

In modelling the cohesive fracture with the finite element method, two main strategies may be followed: discrete inter-element or discrete intra-element cracks. In the first approach, the crack evolves between elements and remeshing operations are required if the crack path is not known in advance [109-111]. An alternative strategy corresponds to model the crack propagation through the finite elements. This intra-element approach includes the incorporation of a discontinuous mode at the element level [112-116]. The use of embedded discontinuities avoids remeshing, enables the use of coarser finite element meshes and simplifies the reproduction of the anisotropy induced by the fracture process. With the intra-element approach, the spatial orientation of the discontinuities depends on the local values of the stress or strain fields. A comparative study of such finite elements with embedded discontinuities may be found in the same reference [117].

Recently, the Extended Finite Element Method [118-124] and the Generalised Finite Element Method [125-131] have been developed to model arbitrary discontinuities in meshes. This extension exploits the partition of unity property of finite elements [132-134], which allows local enrichment functions to be easily incorporated into a finite element approximation. (top)