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This paper proposes a tension-compression damage model for concrete materials, formulated within the framework of thermodynamics of irreversible processes. The aim of this work is to solve the following problems: the premature divergence of numerical solutions under general loading conditions due to the conflict of tensile and compressive damage bounding surfaces, which is a result of the application of the spectral decomposition method to distinguish tension and compression, and the unsatisfactory reproduction of distinct tension-compression behaviors of concrete by strain-driven damage models. The former is solved by the sign of the volumetric deformation, while the latter is solved via two separated dissipation mechanisms. Moreover, of specific interest is an improved solution to the problem of mesh-size dependency using consistent crack bandwidths, which takes into account situations with irregular meshes and arbitrary crack directions in the context of the crack band approach. The performance of the model is validated by the well-documented experimental data. The simplicity and the explicit integration of the constitutive equations render the model well suitable for large-scale computations.

Concrete is still one of the most widely used construction materials in civil engineering applications [

Despite the achievements of CDM over a wide range of materials (e.g., [

The second challenge is the modeling of tension-compression behaviors of concrete materials which exhibit large differences in their peak strengths and postpeak stages, as well as the unilateral effect characterized by a total or partial recovery of the degraded stiffness during the tension-compression load reversal. The classical strain-driven damage models incorporating one damage criterion alone may be incapable of predicting such distinct behaviors. Among them (e.g., [

The third challenge is to avoid the lack of mesh objectivity of numerical results when the modeling of strain-softening is taken into account. At the stage of strain-softening, the concrete does not lose its entire load-carrying capacity but exhibits a progressively decreasing residual strength. Therefore, capturing the strain-softening is important because it is responsible for predicting the behavior of overloaded structures. However, damage models based on conventional continuum theories usually produce mesh-dependent results in FE analyses [

To solve the problem of mesh-size dependency in the framework of continuum mechanics, various types of regularization methods have been developed that can be distinguished by the crack band approach (or fracture energy regularization) [

One key parameter in the crack band approach is the crack bandwidth, through which the fracture energy (regarded to be a material property) is related to the volumetric (or specific) fracture energy, and it therefore ensures an objectivity of the energy dissipation regardless of the mesh refinement. The existing estimations of the crack bandwidth can be divided into three categories: (i) methods based on the square root of the element area (for two-dimensional elements) or the cubic root of the element volume (for three-dimensional elements) (e.g., [

The main purpose of this study is not to describe all the mechanical behaviors of concrete but rather to give methodological efforts to solve the aforementioned difficulties. Therefore, the plastic dissipation is not considered and the isotropic damage is assumed for simplification. Accordingly, the objectives of this paper are fourfold: (i) to define the criterion distinguishing tension and compression by the sign of the volumetric deformation; (ii) to propose a model to describe distinct tension-compression behaviors of concrete materials under a framework similar to that of Cervera et al. [

The thermodynamics of irreversible processes provides a general framework for formulating constitutive equations. In other words, the state of the material can be described by the thermodynamic potential. Consequently, the stresses and other associated variables can be derived from this potential [

To describe the damage due to tension or compression, it first necessitates a criterion that defines tensile/compressive states. As mentioned in Introduction, the method of spectral decomposition may lead to the premature convergence problem in numerical solutions [

The damage criterion characterizes the state of damage and represents a bounding surface whose boundary corresponds to the states at which the damage will grow. To clearly specify the damage due to tension or compression, two damage criteria are introduced as follows:

The damage threshold represents the largest value of the equivalent strain ever reached by the material during the loading history [

Similar to the work of Fichant et al. [

From the macroscopic viewpoint within the framework of thermodynamics, the damage variable can be introduced as an internal state variable to quantify the average material degradation [

In this paper, explicit damage evolution laws are further postulated as follows:

Influence of

At any material point, damage can evolve only if the current state reaches the boundary of the elastic domain, characterized by

The failure of concrete is related to the degradation of physical properties of the material. During this damage process, the energy dissipation is always nonnegative and accompanied by an increase of entropy [

Substitution of equation (

Because of the fact that the inequality of equation (

Using equation (

Therefore, the second principle of thermodynamics is satisfied as long as the damage variables

To solve the problem of mesh-size dependency, a fracture energy regularization based on the crack band approach [

Damage zone and fracture energy regularization: (a) damage zone in the concrete specimen under tensile loading; (b) stress-strain response outside the damage zone; (c) traction-separation law in the damage zone; (d) general stress-strain relation for the concrete specimen.

The crack band approach assumes that the failure of material occurs within a damage zone having a limited width, whereas the other parts are unloaded [

Using equation (

It is noteworthy that equation (

Similarly, the compressive damage threshold

Generally, the crack bandwidth can be regarded as an FE discretization parameter, which depends on the chosen element shape (e.g., quadrilaterals or triangles), element size, element orientation (or mesh alignment) with respect to the direction of the strain localization band, interpolation function or order of the displacement approximation (e.g., linear or quadratic), and numerical integration scheme (number of integration points) [

Inspired by the works of Červenka et al. [

Proposed estimation of crack bandwidths.

For comparison, the existing estimation formulas of the crack bandwidth mentioned in Introduction are recalled as follows.

The tensile crack bandwidth

The tensile crack bandwidth

The tensile crack bandwidth

The geometrical meanings of the aforementioned tensile crack bandwidths and their differences with respect to the direction of

Comparison of estimations of crack bandwidth for a rectangular quadrilateral element: (a) geometrical meanings of crack bandwidths estimated by the present model, Oliver [

Comparison of estimations of crack bandwidth for a right triangular element: (a) geometrical meanings of crack bandwidths estimated by the present model, Oliver [

To establish the validity of the proposed constitutive model, the computed results were either verified theoretically or compared with measured ones from experiments. These are presented in the following sections.

To examine the rationality of the adopted criterion distinguishing tension and compression, and to validate the present model in reproducing the distinct strength features of concrete under biaxial stresses, a series of plain concrete specimens subjected to different stress ratios (denoted by

Biaxial strength envelope of concrete.

Figure

Computed stress-strain curves of concrete under biaxial compression.

To validate the proposed model in capturing the strain-softening, a double-edge notched (DEN) concrete specimen under direct tension, which was investigated experimentally by Hordijk [^{2} in the middle cross-section, was glued to the loading platens and subjected to an axial tensile load under the deformation control. The average deformation ^{2}).

DEN concrete specimen: geometry, boundary, and loading conditions (dimensions in mm).

The material properties of the test specimen used for the numerical analysis were Young’s modulus

The DEN concrete specimen was modeled by a 298-element mesh with four-node plane stress quadrilateral elements using a 2 × 2 Gaussian integration scheme, as shown in Figure

FE mesh for the DEN concrete specimen.

Average stress-deformation curves for the DEN concrete specimen.

To prove the efficiency of the proposed estimation of the crack bandwidth with respect to the FE mesh irregularity, a bar subjected to uniaxial tension was used for the numerical analysis by following the work of Oliver [

A bar subjected to uniaxial tension.

The material properties used for the numerical analysis were Young’s modulus

According to Oliver [

FE meshes used for the study of mesh irregularity (after the study [

The bar was discretized by the four-node quadrilateral or three-node triangular plane stress elements, both of which adopted a linear interpolation function (implying a constant strain field) and a 1-point Gaussian integration scheme. Figure

Due to the fact that the projection methods (e.g., the present method and that of [

Force-displacement responses obtained by different estimations of crack bandwidth with different mesh irregularities: crack bandwidth estimated by (a)

To demonstrate the mesh-size objectivity and the capability of the proposed model in predicting the structural response in shear loading, a single-edge notched (SEN) concrete beam of Arrea and Ingraffea [

SEN concrete beam: geometry, boundary, and loading conditions (dimensions in mm).

The material properties of the SEN beam used in the analysis were Young’s modulus

The SEN concrete beam and the steel plates were discretized by four-node quadrilateral (with 2 × 2 Gaussian integration) and three-node triangular plane stress elements in the FE analysis, respectively. Three different mesh sizes were used for the mesh-size dependency study: coarse, medium, and fine meshes, which had a total of 480, 810, and 1176 elements, respectively (Figure

Different FE mesh sizes for the SEN concrete beam in the mesh-size dependency study: (a) coarse mesh; (b) medium mesh; (c) fine mesh.

Figure

Load-CMSD curves for SEN concrete beam obtained with different FE mesh sizes.

Final tensile damage profiles in the central part of the SEN concrete beam: (a) coarse mesh; (b) medium mesh; (c) fine mesh.

Computed final crack patterns in the central part of the SEN concrete beam: (a) coarse mesh; (b) medium mesh; (c) fine mesh.

According to the numerical results, it is clear that (i) the global structural responses are well predicted compared to the experimental results (Figure

The predicted thicknesses of the macroscopic crack are inconsistent when using different mesh sizes, as shown in Figures

As also can be observed in Figures

This paper presents a tension-compression damage model with consistent crack bandwidths developed for the macroscopic description of concrete behaviors. The model is formulated within the framework of the internal variable theory of thermodynamics so that the constitutive relations are derived without ad hoc assumptions. Based on the numerical and experimental results, the following conclusions can be drawn:

The essential features of the proposed model include the identification of tensile/compressive strain states by the sign of the volumetric deformation, the rational definition of two equivalent strains, two internal variables, and two separated damage criteria describing distinct damage mechanics under tension and compression. Such features enable the present model to provide a better prediction of the distinct tension-compression behaviors of concrete materials. Furthermore, the adopted method using the sign of the volumetric deformation avoids the premature convergence difficult in the numerical analysis, which is often encountered in existing models using the spectral decomposition of a stress or strain tensor.

The main contribution of this paper is that it provides an improved estimation of the crack bandwidth, which is developed for two-dimensional linear elements by taking into account the element shape, element size, and element orientation with respect to crack directions and spatial positions. Its capability in numerical analyses of irregular meshes and arbitrary crack directions has been validated by the comparative study of existing estimations of the crack bandwidth at the element level. The proposed model incorporating this enhanced description of the crack bandwidth has also been proven to be mesh-size independent at the structural level. The present model is able to provide objective simulation results with respect to the finite element mesh refinements that are very important for the computation of the real engineering structures.

The proposed model has been validated by the well-documented experimental data of concrete. The good agreement between the predicted and experimental results consequently demonstrates that the present model provides an effective tool for modeling concrete behaviors in tension and compression.

The present model has a limited number of material parameters, each of which has a clear physical meaning and can be identified by standard laboratory tests. Moreover, the simplicity and the explicit integration of the constitutive equations make the model well suited for large-scale computations.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The work described in this paper was supported by the National Natural Science Foundation of China (51208005), the Transportation Science Research Project of Jiangsu Province (0213Y25), the Natural Science Foundation of Anhui Province of China (1208085QE80), the Educational Department of Anhui Province of China (KJ2012A095), and the Doctoral Fund of Anhui University of Science & Technology (11084).