In this work a two-dimensional formulation describing the fracture process in reinforced concrete is developed, implemented and validated. The cracks in the material are captured by means of continuum strong discontinuity approach (CSDA) (Oliver 1996) and the constitutive model of composite material is defined through mixing theory (Truesdell & Toupin 1960).
The composite material consists of one or two groups of long fibers or steel bars embedded within a concrete matrix. Likewise, each component is characterized by a constitutive model. The concrete is described by a damage model with degradation in tension and compression (Oliver, Cervera et al. 1990). A uniaxial plasticity model (Simó & Hughes 1998) is used for the steel. Also, phenomena as bond-slip and dowel action (Park & Paulay 1975) are included and represented by additional models of interaction between concrete and steel. The initiation and propagation of cracks are understood as a strain localization process described by means of CSDA. A bifurcation analysis of composite material is proposed to establish the bifurcation time and direction of the crack.
The model has been implemented in a two-dimensional analysis program using the finite element method (FEM), where it is assumed material non-linearity and infinitesimal strains. An implicit-explicit integration scheme for the constitutive equation (Oliver, Huespe et al. 2004; Oliver, Huespe et al. 2006) ensures a positive defined stiffness matrix of the problem and increases the robustness and stability of the solution. On the other hand, a strategy to tracking discontinuity paths (Samaniego 2002; Oliver & Huespe 2004), allows that the discontinuity paths correspond among the elements.
According to the proposed formulation, on each point of solid, the strain and stress fields of the reinforced concrete are described as a composite material. This has the following advantages: first, the model facilitates the implementation on the finite element method, since many ingredients of standard numerical process remain, and secondly, the macroscopic scale of analysis avoids the discretization of each component material and the interaction effects, and consequently the computational cost is reduced.
The model can reproduce two different stages of cracking in the reinforced concrete. Initially, the steel capacity and the adherence in the interface produce a stable stage of distributed cracking, where appear many cracks with constant spacing and opening. Afterward, a localization cracking stage is characterized by few cracks while the structural response decreases. Reinforced concrete members subjected to tension, bending and shear are simulated. The numerical results, mainly the structural response and the crack pattern, are compared with experimental test (Leonhardt 1965; Collins, Vecchio et al. 1985; Ouyang & Shah 1994; Ruiz, Elices et al. 1998). The correlation between numerical results using the proposed formulation and actual results is quantitative and qualitatively satisfactory.