Analysis of Fatigue Strain, Fatigue Modulus and Fatigue Damage for the Model Formulation of Concrete Based on Strain Life Approach

Analysis of fatigue strain, fatigue modulus and fatigue damage for the modeling of concrete plays a vital role in the evolution material behaviour which is heterogeneous and anisotropic in nature. The Level-S nonlinear fatigue strain curve, fatigue modulus curve, residual strain curve of concrete in compression, tension, flexure and torsional fatigue loading were proposed using strain life approach. The parameters such as physical meaning, the ranges, and the impact on the shape of the curve were discussed. Then, the evolution model of fatigue modulus was established based on the fatigue strain, fatigue modulus, residual strain and secondary strain evolution model. The hypothesis of fatigue modulus is inversely related with the fatigue strain amplitude. The fatigue evolution of concrete damages the bond between material grains, changed the orientation of structure of molecules and affects the elastic properties resulting in the reduction of material stiffness and modulus by formation of microcracking, macro cracking, cracking and finally damage. This paper presents the fatigue strain life model and analysis of fatigue strain, fatigue modulus and damage parameters of concrete which is capable of predicting stiffness degradation, inelastic deformation, and strength reduction under fatigue loading and experimental results were employed for the validation of the theoretical model.


Introduction
In recent years, concrete has been widely used in every civil engineering works form between maximum strain rates with respect to number of load cycle for the second phase of the concrete. Under the compression fatigue loads, the form was a linear relationship. [10] also found out the index formula for fitting of second phase of concrete regarding fatigue strain. [11] conducted the compressive fatigue strain experiment for the achievement of two-staged nonlinear formula. On the base of fatigue strain revolution different methods of analysis showed much more deficiencies. Till date, the linear three-stage fatigue equations are described in good order of low. The three stage nonlinear equation is of high precision showed the complicated form. Due to lack of whole fatigue strain curve with cycle relationship, the fitting of fatigue strain curve with cyclic load relationship is advanced in recent researches. There have been few researches carried out in fatigue strain for conditions in which the fatigue stress is smaller than limited stress but greater than the threshold value n. A discussion has reached the variation law of the three-stage fatigue strain, and some different experienced curve fitting equations have been obtained, but the initial strain is not taken into consideration. The significance of factor related to fatigue strain is not clear enough which results the unstable fittings of coefficients.
The concrete is a complex hybrid composite material. During cyclic loading fracture may be occurred by fracturing of the cement paste, fracturing of the aggregate, failure of bond between the cement pastes aggregate or any combination of these mechanisms. It contains voids and microcracks at the initial phase of preparation due to workability of concrete in presence of water before applying load. The fracture arised during cyclic loading due to increase in the stress concentration at the points of voids and microcracks. Forces that are required to obtain the fracture are usually much less than forces that would have been required in the case of monotonic loading. Fatigue is a process of damaging materials due to progressive permanent internal changes in the materials that occur under the actions of cyclic loadings. These phenomena can cause progressive growth of cracks present in the concrete molecular system and eventually fail structures when high levels of cyclic loads applied for shorter period of time or lower levels of loads are applied for longer period's times. This causes the failure of structure made of concrete with value lesser than the ultimate strength of the concrete. The Concrete, a heterogeneous material comprised of the mixture of cement, sand and aggregates, exhibits several mutually interacting inelastic mechanisms such as micro crack growth and inelastic flow even under small magnitude of cyclic load applied in large number of cycles. As a result, the concrete material does not guarantee endurance fatigue limit like metal as described by Miner's hypothesis.
The reinforced concrete structures such as bridges, hydraulic foundations, pressure vessels, crane beams are generally subjected to long term cyclic loading.
The fatigue in concrete was experienced rather late, with comparison to steel.
The fatigue resistance of the concrete is influenced by many different factors e.g.
proportions in mixing ratios of manufacturing concrete, original moisture content, aggregate size, quality of sand, and load effects such as load frequency, na- of micro-cracks is due to thermal strain, which is caused by temperature variation. When the micro-cracks propagate the fatigue process starts. At the beginning of the loading the propagation of the micro-cracks drives slowly. As loading continues the micro-cracks will proceed propagation and lead to macro-cracks, which may grow further. The macro-cracks in failure nature determine the remaining fatigue life caused by stress until failure occurs.
Depending on the range of the number of cycles to which a structure is sub- when applies for large number of cycles, the concrete does not guarantee endurance fatigue limit like metal as described in [12]. The presence of permanent damage at fatigue failure has been documented by a number of investigations. [13] developed fatigue damage model for ordinary concrete subjected to cyclic compression based on mechanics of composite materials utilizing the concept of dual nature of fatigue damage, which are cycle dependent and time dependent damage. The formulated model is capable of capturing the cyclic behavior of plain concrete due to progressive fatigue strain with the increase in number of loading cycles. [14] used accelerated pavement testing results to carry out cumulative fatigue damage analysis of concrete pavement. He reported that the Miner hypothesis does not accurately predict cumulative fatigue damage in concrete. The experimental work of [15] clearly showed that the increase of damage in the material takes place about last 20% of its fatigue life. [16] presented a theoretical model to predict the fatigue process of concrete in alternate tension-compression fatigue loading using double bounding surface approach described in strain-energy release rate by constructing the damage-effective tensor.
In the past few years, a number of damage constitutive models have been published to model the observed mechanical behavior of concrete under monotonic and cyclic loading [16]- [22]. The constitutive models well describe the design of structural life arose from the physical observation of two dominant microstructural patterns of deformation in concrete having inelastic flow and microcrack-

Fatigue Strain Evolution Model
Depending upon the different stress types, three-stage variation law of fatigue evolution model was proposed. Moreover, some valuable physical parameters like initial strain, instability speed of the third stage as a form of acceleration which is directly proportional to the total fatigue life of concrete. Mathematically, the model could be obtained as below In formula (1), 0 ε = initial strain and n ε = fatigue strain, n = cycle times of fatigue loads, Equation (2) represents a formula for maximum strain and Equation (3) represents the formula for the residual strain.
On the basis of the elastic proportional limit, if the upper limit of fatigue stress is larger then fatigue strain increases quickly. The slope of the curve regarding this increment will be larger and become vertical that causes the degeneration of the three-stage curve. When the upper limit of fatigue does not exceed the threshold value, the elastic strain should be added to the initial strain and value became unchanged, which shows similarity in the curve formulation. With the experimentation, it can be shown that the value of most stresses falls in between the value of threshold and upper limit.
Being the maximum and minimum value of stress and strain in fatigue test, two types of the curve regarding maximum strain i.e. 0 max ε and residual strain i.e. 0 res ε with respect to the cyclic number are obtained. The main causes for obtaining these two types of curve are due to defects in materials and preloading conditions. It is very much difficult to differentiate these two maximum and residual value, so experimentation of fatigue test has become essential.
Therefore, when the fatigue loading reaches to the upper limit then, the corresponding maximum strain 1 max ε and residual strain 1 res ε are obtained which is adopted in this paper. For comparison, the obtained strain 1 max ε and 1 res ε compared to the actual experimental data i.e.
( ) In this formula, unstable ε is a total strain of concrete in an unstable state. For the study of fatigue strain parameters , α β and p, on the basis of evolution law of fatigue strain curves, divided by fatigue strain in both side of formulas (4) and (5), we get Equations (4) and (5)  β = destabilizing factor the value of which depends on p and α . If n/N f (Circulation ratio) is equal to 1, the coordinate point (1, 1) will be adopted in formulas (4) and (5), thus obtained the values of β as formula (6) and (7), which is the maximum fatigue strain and the residual fatigue strain.
From Equation (4)  It is obviously shown that the rate of convergence speed of p influences the convergence speed of curve in S nonlinear model. The instability speed factor increased at the faster increment of p when the third stage of the curve will grow faster. Therefore, the factor p should be located in the curve.

Fatigue Modulus Evolution Model
Mathematically Fatigue Modulus i.e. "E" is defined as, where, max strain decreases. The strain evolution curve is the level-S-shaped curve reversed as the curve of fatigue modulus shown in Figure 4. Suppose two curves are symmetrical about the straight line, parallel to X-axis, y = D, and the point in the fatigue strain curve is (x, y). Therefore, corresponding point on the curve of the fatigue modulus is (x, 2D − y). Figure 4 illustrates the best Model regarding Damage Modulus evolution Model. Therefore, the normalized fatigue modulus can be expressed as: The mathematical formula for fatigue modulus evaluation model is

Fatigue Damage Evolution Model
Before performing the fatigue test experiment, preloading test should be done to realize the upper limit of fatigue load which creates the damage. Initial damage is categorized in two parts, i.e. damage caused by the material defects and the damage caused by application of load. After achievement of clear concept of above said mechanism, damage regarding maximum fatigue strain during shock and vibration can be written as; where, 0 = initial damage. Adopting the fatigue strain evolution formula from Equation (2) into Equation (11), the damage evolution equation based on the maximum fatigue strain is developed. The damage evolution formula is as under: Similarly, by definition, damage based on fatigue residual strain can be expressed as under: Putting the value of fatigue residual strain evolution formula from Equation (3) into Equation (14), the damage evolution equation based on the fatigue residual strain is generated. The expression is as follows: For data fittings, if It is clearly shown that at the increment of concrete damage circulation ratio, the concrete damage grows rapidly to the middle stage linearly which is similar to strain evolution, it has at wastage variation as shown in Figure 5.

Fatigue Damage Model in Strain Space by Using the General Framework of Internal Variable Theory of Continuum Thermodynamics
For isothermal process, rate independent behavior and small deformations, the Helmholtz Free Energy (HFE) per unit volume can be deduced as follows: :  Utilizing fourth order material's stiffness tensors, the stress regarding damage process is given by mathematical relations: The first derivative form of Equationn (18) with respect to cyclic number N is given by where e σ is the stress increment in the absence of further damage in the material, D σ is the rate of stress-relaxation due to further microcracking (elastic damage), and ( ) i k σ designates the rate of stress tensor to the solid body which is irrecoverable or permanent deformation due to microcracking.
The assumption is made such that damage during fatigue loading alters elastic properties and affects the stiffness tensor. For small deformation, the following decomposition of the fourth-order stiffness tensor, E, is adopted. where where L and M are fourth and second order response tensors that determine the directions of the elastic and inelastic damage processes. Following the Clausius-Duhem inequality, utilizing the standard thermodynamic arguments and assuming that the unloading is an elastic process, a potential function : : Equation (6) is obtained in damage surface which establishes the onset of material inelasticity and stiffness reduction. In Equation (22), is interpreted as the damage function given below as for some scalar valued function The substitution of response tensors L and M from Equations (24) and (25) into Equation (26)   The damage function p(k) obtained from uniaxial tensile test for concrete materials based on the experimental results of (24) is given in (24) as And for elastic damaging process, ( 0 β = ), the limit damage surface reduces where u ε represents the strain corresponding to uniaxial tensile strength of concrete, which is used as the reference strain and hence the result of a conventional uniaxial tensile test is needed to establish u ε .
The behaviour of concrete material due to fatigue strain applying goes progressive permanent structural change and the material fails at stresses having a maximum value less than the tensile strength of the material. In this paper, it is assumed that within the damage surface of the given strain state, the unloading-reloading cycles (Fatigue loadings), increase amount of damage of concrete due to the growth of microcracks leading to inelastic deformations and stiffness degradation, which eventually reduces the ultimate molecular strength of the concrete. To achieve this, the damage surface is modified to predict the increase in damage in the material with increasing number of cycles of loading as where X(N) is a function that depends on the number of loading cycles and adopted to increase of damage with increasing number of cycles. We propose a power function for X(N) as Here, N represents the number of loading cycles, and A is a material parameter. Utilizing Equations (28) Finally, the rate of damage parameter, k  , must be used in the simple constitutive relation of the form given by Equation (33) in uniaxial tensile stress path for representing inelastic deformation, stiffness degradation and strength reduction due to fatigue strain. The corresponding equations be ( ) : : Equations (34)  1 : 2 exp : where,   Table 1 and Table 2.  Figure   17]. For numerical simulation, the following constant were used, A = 0.10 and β = 0.15 and 0.00 in two cases, parameter A is estimated by comparing predicted results and experimental results over a range of applied strains from Table 2 and   Table 3.

Damage Linear Model to Estimate the Life of Plain Concrete in Compression, Tension and Flexure
It is widely accepted that the [23] and [24] can be used to         relationship between the fatigue strength of concrete after a given number of cycles and the ratio of the minimum to maximum stress level. The material parameter (β) required in the model proposed by [23] and [24] was 0.064. [25] and [26] proposed a material parameter of 0.069, after conducted flexural tests on plain concrete. To account for other plain lightweight concrete, [26] and [27] proposed a material parameter of 0.0685.
In Equation (35), S max = ratio of the maximum stress level to the concrete compressive strength, N f = number of cycles to failure, And R = stress ratio [minimum stress level (σ min ) to maximum stress level (σ max )]. As a means of predicting the fatigue life of steel fiber concrete using the [23] and [24] stress-life model, [23] and [24] developed material parameters for steel fiber concrete with fiber volume contents of 0.5%, 1%, and 1.  attempt seems to be reasonable, the material parameters obtained indicated a lower fatigue life as the steel fiber volume increases from 1.0% to 1.5%. The proposed material parameters also exhibited higher fatigue life for steel fiber concrete with volume content of 0.5% compared with steel fiber concrete with a volume content of 1.5%. where, ε sec = secondary strain rate, and "A" and "B" are constants to be obtained from experiments. A power equations correlation (nonlinear) was initially proposed by [26] and [27]; however, the correlation between the number of cycles resulting to fatigue failure and the secondary strain rate is expressed in a linear form Equation (38) to simplify the analysis required for deriving material parameter "β" for plain concrete and steel fiber also.
A relationship between log(1 − S max ) the secondary strain rate (ε sec ) can be given thus: "C" and "D" are constants obtained from experiments as given in Table 4.

Degradation in Residual Strength and Secant Modulus
Modified damage evolution models has been proposed for concrete residual strength and fatigue secant modulus [Equation (44)] [7]. The models are functions of the maximum stress level, critical damage value, loading frequency, damage parameter "s" and material parameter β. The critical damage is the percentage reduction in concrete strength or fatigue secant modulus at failure. The mathematical Model is From Zhang et al. (1996), on influence of loading frequency  where, N = number of cycles; s = damage parameter; D cr = critical damage value; and C f accounts for fatigue frequency.
From different investigation reports in the literatures have shown that the concrete strength and the stiffness reduce progressively after fatigue loading cycles have been applied [28]- [34]. To account for this, concrete strength may be modified by using a damage factor D fc . Similarly, the stiffness or secant modulus of concrete may be modified using a corresponding damage factor D ce . Models used for these cases are given thus where,

Irreversible Strain Accumulation Model
Compressive and tensile plastic strains accumulate in concrete under fatigue loading [28]- [34]. However, the magnitude in tension is usually small, and it is Based on an experimental investigation conducted on the strain evolution of concrete in compression [33] and [34] models were proposed for the irreversible fatigue strain (d) as follows: (N f is the number of cycles to failure and N is the fatigue loading cycles) where, 0 d ε is the strain due to loops centerlines convergence,

Results and Discussion
Experimental Study: In the first group of fatigue tests, 22 specimens were tested at different numbers of cycles at a constant stress level and subsequently subjected to monotonic loading. The obtained stress-strain curves from monotonic loading were plotted alongside the stress-strain envelope to observe the intersection of the peak stress of the stress-strain curves with the softening portion of the stress-strain envelope.
In the second group of fatigue tests, 16  Preparation of Test Specimen: The concrete specimens were made from Portland cement, sand, and limestone aggregates (10 mm maximum size) with three different mix ratios. The concrete from the first two batches (Table 4) was cast using a mix proportion of 1:2:2 with a water/cement ratio of 0.5, indicating cement, sand, and coarse aggregate by weight respectively. Mix proportion ratios of 1:2:3 with a water/cement ratio of 0.5 and 1:2:4 with a water/cement ratio of 0.6 were used for the third and fourth batches respectively. The static strengths of concrete after curing for 28 days were obtained for each batch (Table 4), while the fatigue tests were conducted 30 to 40 days after casting.
Percentages of the average compressive strengths of the four batches (69% to 80%) were used as maximum stress levels for the fatigue tests conducted on 16 specimens to failure ( Table 5). The 22 specimens loaded to different numbers of cycles less than the number of cycles leading to failure at a constant maximum stress level of 0.74 (Table 5) and a frequency of 5 Hz are given in Table 5. A constant minimum load of 5 kN was used for all the tests conducted. The approach for estimating the fatigue secant modulus will be discussed in a subsequent section.
The progressive strain readings of the concrete cylinders (100 mm diameter x 200 mm height) tested under uniaxial constant fatigue loading in compression were obtained using a data acquisition system. The specimens tested to failure were used to verify [24] [25] [26] and [27] assumption and that of the intersection of the hysteresis loop with the stress-strain envelope at failure. The 22 curves generated were used as the stress-strain envelopes required to verify the intersection of the peak stresses for the statically loaded fatigue-damaged specimens. However, the stress-strain curves obtained from the experiments were also included in the plots.
Failure Modes: Figure 18 shows specimens in various damaged states. The hairline cracks parallel to the applied loading direction were initially observed in the entire specimen. Thereafter, the cracks widened and finally failed in the form of faults as per Figure 18 and Table 6.

Fatigue Damage Model by Utilizing Continuum Thermodynamics Approach
Strain Based Model of Concreet Material during low frequency is presented in this paper by utilizing the framework of continuum thermodynamics of Continuum Mechanics by taking two material fatigue damage parameter i.e. mathematical symbol, A and β (where, A = fatigue damage parameter regarding energy microcracks of the material particle and another is β = kinematic damage parameter i.e. phenomena of material crack surface close perfectly after unloading).
Fatigue damage evolution law together with the damage response functions was used in the constitutive relation to demonstrating the capability of the model in capturing the essential features of concrete material, such as stiffness reductions, increase in damage parameters, behaviour of fatigue modulus and the inelastic deformations, under fatigue loading environment by finding out the cumulative Engineering fatigue damage parameter i.e. symbolically, K. The fatigue curve relavant to A = 0.10 and β = 0.15 and 0.00 is generated by the modeling and after that this generated model curve is compared to the curve obtained by [16] which shows similar tread of generation of fatigue curve. This also shows the good relation between results obtained from modeling and experiments. Lower value in the experimental curve is due to 0.85 times maximum stress level whereas, modeling takes 100% value.