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An anisotropic damage mechanics model is presented to describe the behavior and failure of concrete under biaxial fatigue loading. Utilizing the approach of bounding surfaces, the limit surface becomes a special case when the number of loading cycles is set to one. By increasing the number of loading cycles, the strength of concrete gradually decreases and the limit surface is allowed to contract and form new curves representing residual strengths. The magnitude of loading, load range, and the load path are known to influence the fatigue life and hence are addressed in this formulation. In this paper, a strength softening function is proposed in order to address the reduction in the strength of concrete due to fatigue. Separate softening functions are also proposed to account for the deformation characteristics in concrete under cyclic loading. Numerical simulations predicted by the model in both uniaxial and biaxial stress paths show a good correlation with the experimental data available in the literature.

The fatigue behavior of concrete has received a considerable attention among researchers in the past two de- cades. This can be attributed to the increasing use of concrete as a construction material. Concrete has been used in various structures due to its unique features such as high compressive strength, good resistance to aggressive and moist environments compare to some other construction materials, and enhancement in strength and deformation capacity under confining stresses. Concrete has been used in dams, bridges, and highway pavements in which cyclic loading is considered as one of the factors affecting its mechanical behavior during its service life. Various research studies have been published on the effects of fatigue loading on the mechanical behavior of concrete in terms of strength, deformation characterization, and modulus of elasticity. Most of these studies were conducted on the uniaxial loading of the material [

It is generally accepted that concrete under cyclic loading loses its strength gradually with an increase in the number of load cycles regardless of the loading path (uniaxial or biaxial). The strength loss during the fatigue process is due to nucleation and propagation of microcracks. During cyclic loading, these microcracks increase and grow to a stage in which major cracks are formed and reduce the load carrying area tremendously. At that point the strength of the material is decreased substantially and approaches the amplitude of the cyclic loading. This results in sudden rupture. It has been argued that at any given cycle, the fatigue strength of concrete under biaxial compression is greater than that under uniaxial compression [

In addition to the strength reduction, fatigue loading affects the modulus of elasticity and the deformational capacity of concrete as well. Awad [

According to Gao and Hsu [

Realizing the fact that fatigue loading has a significant influence on concrete serviceability and may lead to an abrupt material failure, an accurate and efficient model which could capture the behavior of concrete is needed.

In this paper, an approach based on continuum damage mechanics is proposed to model the behavior of concrete under fatigue loading. The general theory of bounding surface approach proposed by Wen et al. [

The general formulation shown in the following is based on the damage mechanics approach and follows the framework of the internal variable theory of thermodynamics. For isothermal and small deformations, the Gibbs Free Energy is obtained as follows [

where C is the compliance tensor,

where ε represents strain tensor. The compliance tensor, C, is assumed to take an additive decomposition form as:

where C^{0} and C^{c} are the initial undamaged compliance tensor of the material and the added flexibility tensor associated with the accumulation of damage, respectively. Due to the nonlinearity behavior between stress and strain for brittle materials, the rate form of the flexibility tensor must be considered as:

In Equation (4), the response tensor, R, determines the direction at which damage occurs. For isothermal and small deformation, the internal dissipation inequality can be represented by Gibbs Free Energy as:

It is also assumed that the damage is an irreversible phenomenon in which,

where

Guided by the experimental data [

where f_{c} is the compressive strength of concrete, E_{0} is the initial stiffness, and “e” represents the natural number. In this paper only the compression mode of damage is considered. Guided by work of Wen et al. [

where “⊗” is the tensor product operator,

The bounding surface approach for fatigue was proposed by Wen et al. [

In order to capture the described behavior of concrete under cyclic loading, an evolutionary equation is needed to predict the failure surface. To accomplish this task, the damage function is restructured to be the product of two functions as shown below:

where

function,

By considering a fatigue uniaxial compression path and substituting Equation (9) into Equation (6), the following form is obtained for the softening function:

where σ is the residual strength of the concrete after specific number of cyclic loading. Equation (10) is a representation of so-called S-n curves. Based on the researches reported in [_{max}; stress ratio, r; and finally the load path all contribute to the fatigue life of concrete. While the fatigue life of concrete is adversely affected by the amplitude of loading, Aas-Jakobsen and Lenschow [

where n is the number of cyclic loading and A and B are material parameters. Utilizing this softening function and incorporating it into the Equation (6), residual strength surfaces could be obtained under various load paths. The inclusion of the first and second invariants of the stress tensor allows the formulation to model load path dependency observed in fatigue testing.

In _{max} and the fatigue failure strain in uniaxial compression is given by

To fully describe the stress-strain behavior of concrete under fatigue loading, four factors including reduction in strength, increase in ultimate strain, plastic strain after each cycle, and reduction in modulus of elasticity need to be addressed. The reduction in strength has been already addressed by the strength softening function; Equation (11). For deformation, as was discussed earlier, concrete under fatigue loading fails at an ultimate strain greater than the one under monotonic loading state. Awad [

where _{u} is the ultimate strain under monotonic loading, and

In this section, results predicted by the model are compared with the experimental data obtained from literature. Material parameters α, A, B, β, and γ are calculated based on the experimental data presented.

Figures 4-6 show the strength versus number of loading cycles for concrete under cyclic uniaxial and biaxial paths with stress ratios of 0.5 and 1.0. These figures show that the strength of concrete materials would decrease with increase in the number of cycles, n. The rate of strength reduction for these three figures are different, meaning that the strength loss is also dependent on the load path. This is consistent with the experimental data and is captured by the proposed model. For Figures 4-6, the following material parameters are used: α = 0.745, A = ‒0.0431, and B = 0.552.

which is in consistent with the experimental data in the literature. For the following figures, the material parameters used are: α = 0.94, A = ‒0.0263, β = 0.0787, and γ = 0.1241.

loading with amplitudes of 0.95f_{c} and 0.9f_{c}. The reduction in strength and longitudinal modulus and increase in ultimate strain are predicted by the model. It can also be noticed that the ultimate and residual strain predicted by the model for fatigue loading with 0.9f_{c} amplitude is greater than the ones for 0.95f_{c} that follows the arguments discussed earlier in the paper. Not all of the cycles to failure are shown in

An anisotropic model is utilized to predict the strength behavior of concrete under biaxial compressive fatigue

loading. Under cyclic fatigue loading, the limit surface is allowed to contract and form new surfaces identified as residual strength surface. This is accomplished by proposing a softening function that is based on amplitude, stress ratio, and load path. By including these parameters, the effects of strange range and the load paths on the fatigue life of concrete are studied and predicted. Furthermore, to capture the effects of fatigue loading on stress-strain behavior of concrete, two additional strain softening functions are proposed for changes in ultimate and residual (plastic) strains. The influencing factors on ultimate and plastic strains such as amplitude, load path, and load range are incorporated into the proposed softening functions. At the end, the results obtained from the model are compared with the experimental data in the literature showing a good comparison.

This research was supported by a grant from US DOT to the Department of Civil and Environmental Engineering at North Dakota State University through MPS program. The authors would like to thank US DOT for the financial support for conducting this research.