Biaxial Fatigue Bounding Surface Concrete Damage Mechanics Stress-Strain
1. Introduction

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  - , while only a few studies could be found in the literature on the effects of biaxial stress state loading  - .

According to Gao and Hsu  the fatigue strain of concrete is comprised of three parts: irreversible strain caused by cyclic creep under the action of average stresses; irreversible strain caused by fatigue cracks; and fatigue strain range. In the same study  , it was reported that the modulus of elasticity of concrete degrades during fatigue process due to damage accumulation which happens as a result of microcracking.

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.

2. General Formulation

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, is the stress tensor, k is a scalar damage parameter, and is a scalar function associated with the surface energy of microcracks. The symbol “:” represents a tensor contraction operation. A constitutive relation for concrete like materials is used as:

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

where C0 and Cc 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,. By combining Equations (1) through (5), the general form of the damage surface is given by:

where is called the damage function. The condition describes the elastic condition for the material which is enclosed by the damage surface. The condition is not allowed for rate-independent processes.

Guided by the experimental data  , the following form for the damage function is postulated as:

where fc is the compressive strength of concrete, E0 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.  and Saboori et al.  , the damage mode is identified by the response tensor R given as:

where “⊗” is the tensor product operator, represents the negative cone of the stress tensor, is defined as the Heaviside function of the maximum eigenvalue of, and I and i are the fourth and second order identity tensors, respectively. The material parameter, , shown in Equation (8) is a strength rated parameter and can be obtained by a biaxial monotonic loading test.

3. Bounding Surface Approach

The bounding surface approach for fatigue was proposed by Wen et al.  in order to predict the behavior of woven fabric composites under fatigue loading. This surface is shown schematically in Figure 1. In the case of fatigue loading, as cyclic loading is applied, the limit surface is allowed to contract and to form residual strength curves. This reduction in strength is caused by damage and microcracks generated during the fatigue process. As the number of load cycles increases, the strength continues to decrease further and the residual surfaces also shrink. The reduction in strength continues to a point at which the residual strength becomes equal to the magnitude of loading. At this point, failure surface is formed and the material cannot withstand any additional cycles resulting in failure.

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 is regarded as the strength softening function. The number of cycles of loading to failure is given by “n” and “r” is the stress ratio (ratio of minimum stress to maximum stress). The dependency of the

Schematic representation of bounding surfaces in biaxial stress space

function, , on “n” and “r” is supported by the experimental observation described in the previous section.

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    , amplitude of loading, σmax; 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  reported that increasing stress ratio results in a greater fatigue life at a given stress. Moreover, considering the data provided by Yin and Hsu  , it is apparent that the rate of reduction in concrete strength is not the same for different load paths. Guided by these findings, the following softening function is proposed in this paper as:

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 Figure 2, a schematic representation of stress-strain behavior of concrete which is consistent with the experimental data  is illustrated. The applied stress is signified as σmax and the fatigue failure strain in uniaxial compression is given by. The figure shows the reduction in strength due to fatigue while the failure strain increases under cyclic loading compared to monotonic loading state.

4. Numerical Example

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.

Figure 3 shows the prediction results of residual strength surfaces in biaxial stress space against experimental data work of Nelson et al.  . The damage surfaces show a good correlation for monotonic loading when n = 1 as well as for fatigue loading when n = 10, 100, and 1000 with experimental data. For Figure 3, following material parameters are used: α = 0.587, A = ‒0.0445, and B = 1.521.

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.

Figure 7 illustrates the comparison between the experimental data provided by Awad  and S-n curves obtained by the model for three uniaxial fatigue loading with different stress ratios. As shown, this model captures the effect of stress range on fatigue life of concrete. It can be seen that for any constant stress amplitude, the model predicts a greater fatigue life for a stress range of 0.65 - 0.68 than a stress range of 0.41 - 0.47 and 0

Residual strength surfaces for various number of cyclic loading, experimental data by Neslon et al. [<xref ref-type="bibr" rid="scirp.53842-ref11">11</xref>] S-n curve for concrete under uniaxial cyclic loading, experimental data by Yin and Hsu [<xref ref-type="bibr" rid="scirp.53842-ref13">13</xref>] S-n curve for concrete under biaxial cyclic loading with stress ratio 0.5, experimental data byYin and Hsu [<xref ref-type="bibr" rid="scirp.53842-ref13">13</xref>]

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.

Figure 8 and Figure 9 represent the capability of the model in predicting the ultimate strain and residual strain of concrete under uniaxial fatigue loading after different cycles of loading. The model predicts higher ran- ge of ultimate and residual strain for fatigue loading with lower amplitude. This is in conformity with the discussion that was presented earlier and implies that concrete becomes more flexible under fatigue loading with lower amplitude. Also, Figure 9 shows the increase in residual strain by increasing the stress range. That is, for a given “n”, the residual strain increases with increasing stress range.

Figure 10 and Figure 11 show the stress-strain curves of concrete under uniaxial monotonic and fatigue

S-n curve for concrete under biaxial cyclic loading with stress ratio 1.0, experimental data by Yin and Hsu [<xref ref-type="bibr" rid="scirp.53842-ref13">13</xref>] S-n curves for concrete under uniaxial loading with various stress ranges, experimental data by Awad [<xref ref-type="bibr" rid="scirp.53842-ref4">4</xref>] Ultimate strain versus number of cycles for concrete under uniaxial cyclic loading, experimental data by Awad [<xref ref-type="bibr" rid="scirp.53842-ref4">4</xref>]

loading with amplitudes of 0.95fc and 0.9fc. 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 0.9fc amplitude is greater than the ones for 0.95fc that follows the arguments discussed earlier in the paper. Not all of the cycles to failure are shown in Figure 11 for clarity.

5. Conclusion

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