Algorithm for determina on of the damage characteris c (C-S) curve of asphalt mixtures

Recebido: 6 de setembro de 2019 Aceito para publicação: 19 de novembro de 2019 Publicado: 15 de maio de 2020 Editor de área: Kamilla Vasconcelos ABSTRACT This paper presents the algorithm for calcula ng the damage characteris c curve obtained in direct tension tests taking into account sinusoidal controlled strain loading. The Viscoelas c Con nuum Damage formula on is presented in a summarized form for the algorithm, in which the pseudo strain, at the instants associated to the observed stress, is calculated using the expression of the linear viscoelas city stress under controlled strain tes ng. This facilitates subsequent treatment of the data to obtain the vs. curve. The proposed algorithm is simple to understand and easy to implement computa onally. The algorithm was validated with the results of fa gue test simula ons in three mixtures, which have indicated its poten al.


INTRODUCTION
Fatigue cracking is one of the major distresses in asphalt pavements. The cause of these cracks, which are affected by repeated (i.e., cyclic) loading over time, can be tied to aspects such as weak pavement sublayers, insuf iciently designed asphalt materials, or changes in strain tolerance of the mixture due to ield aging.
There are different approaches to quantify the fatigue crack process in asphalt mixtures. Among the different approaches, Nascimento (2015) proposed the use of the Viscoelastic Continuum Damage (VECD) for characterizing Brazilian asphalt mixes in terms of fatigue cracking. The use of the continuum damage approach to study fatigue cracking in asphalt mixtures was initiated by Kim and Little (1990), who presented the behavior of cyclic tests on a prismatic sample as well the bene its of rest periods. Continuum damage mechanics ignores speci ic microscale details and attempts to characterize the material by considering the effect of microstructural changes on observable macro properties (Lemaitre, 1996). It considers the effect of damage in the constitutive modeling of viscoelastic materials by quantifying two variables: the damage parameter ( ) and the pseudo stiffness ( ). The damage parameter, in general, quantiies any change in the microstructure that results in a reduction of stiffness.
Fatigue curve is a characteristic of asphaltic materials that can be used to predict the durability of asphalt surface courses through computer simulation. It is obtained through laboratory fatigue tests, which measure the integrity of the specimen, represented by the stiffness value over time. Kim (2009) developed an algorithm for adjusting the damage characteristic curve (i.e., the vs. curve) of asphalt mixtures from the stresses observed in controlled strain tests performed on laboratory specimens.
The algorithm presented herein is an improvement of the referred algorithm. The original algorithm proposes to arbitrate the increment of damage, while the new algorithm proposes to arbitrate the initial damage . The original algorithm calculates the rate of change of integrity with damage growth, incrementally, whereas the proposed algorithm calculates through the derivative of the equation that relates integrity with damage parameter. The proposed algorithm calculates the pseudo-strains from the LVE theoretical stresses due to the applied strains. This paper is organized as follows: Section 2 presents the theoretical basis of formulation; Section 3 presents the proposed algorithm; Section 4 shows results of three mixtures analyzed with the proposed algorithm; and Section 5 presents the conclusions.

THEORETICAL BASIS
The damage characteristic curve represents the fundamental relationship between damage and pseudo stiffness (material integrity) for asphalt mixtures. Combined with the linear viscoelastic properties of asphalt concrete, the damage characteristic curve can be used to analyze the fatigue characteristics. This section presents a brief theoretical basis for obtaining the damage characteristic curve.

Linear Viscoelas2city
Viscoelastic materials are those for which the relationship between stress ( ) and strain ( ) depends on time ( ). Linear viscoelasticity is a theory describing the behavior of such ideal materials. Linear viscoelastic (LVE) materials exhibit time and temperature dependent behavior, and their current response is dependent on both the current input and all past input (i.e. the input history). By contrast, the response of an elastic material is only dependent on the current input. Constitutive relationship for LVE materials is typically expressed in the convolution integral form, as follows: where is the relaxation modulus. The analytical forms for this function is generally given by Prony series: where: , are stiffness constants and are relation times. The formulation through Prony series has the advantage of being linked to the behavior of the Maxwell-Wiechert mechanical model as shown in Figure 1. Note that the characterization of LVE behavior is undertaken by performing temperature and frequency sweep tests which are then processed to obtain the series coef icients.

Elas2c-Viscoelas2c Correspondence Principle in Terms of Pseudo-Variables
The theory of linear viscoelasticity and some nonlinear viscoelasticity models have been established in the past (Pagen, 1965;Ferry, 1980;Lamborn and Schapery, 1993;Park et al., 1996). However, solution of various kinds of viscoelastic boundary value problems still remains a complex problem. Fortunately, the nature of the correspondence existing between the elastic and the integral-transformed (e.g. Laplace-transformed or Fourier-transformed) viscoelastic ield equations for linear non-aging materials enables one to obtain a solution to a viscoelastic boundary value problem from the solution of its corresponding elastic boundary value problem (Park et al., 1996).
Originally proposed by Schapery (1984), the elastic-viscoelastic correspondence principle in terms of pseudo-variables involves the use of parameters which do not necessarily represent physical quantities such as strains and stresses, but which are useful in transforming the viscoelastic stress-strain relationships into elastic-like equations.
On the other hand, it has been shown that the stress-strain relations for a broader class of viscoelastic materials can be represented by elastic-like equations through the use of so-called pseudo-variables. This simplifying feature enables correspondence principles to be established and applied to linear and nonlinear viscoelastic boundary value problems, with or without aging (Schapery, 1981;Schapery, 1984). The Laplace or Fourier transform are not used in this approach.
Using this correspondence principle one can obtain viscoelastic solutions from their elastic counterparts through a simple substitution and integration. This principle is applicable to stationary or time-dependent boundary conditions and does not requires a transform inversion step, but it rather requires only the evaluation of a convolution integral (Kim, 2009;Nascimento, 2015).
The constitutive law for a viscoelastic material can be expressed in a form similar to Hooke's law of linear elasticity for all times. Using pseudo strain in place of physical strain, the uniaxial constitutive relationship presented in Equation (1) can be rewritten as follows: (3) where ! is a particular reference modulus, typically taken as one, and ! is the pseudo strain: Therefore, pseudo strain can be de ined as the numerical value of the stress (without damage) divided by ! .

Work Poten2al Theory
The Work Potential Theory (WPT) damage model is based on linear viscoelastic characterization and nonlinear damage accumulation. Schapery's WPT Theory (Schapery, 1994) contains parameters that characterize the variation of material integrity with damage growth. To clarify the physical signi icance of these parameters, a step-by-step process is followed to link the damage functions to the parameters.
The pseudo strain energy density function of the material is given by: where the pseudo stiffness is a function of a damage parameter S. Then, stress is obtained from derivation of the pseudo strain energy: According to Schapery (1994), the following form can be used to describe damage evolution in a viscoelastic material: where ( is the damage evolution rate over time, $ ! is the pseudo strain energy density function, and α is a material constant. According to Park et al. (1996), the constant α is based on LVE fracture mechanics. Its value is related to the viscoelastic characteristics of the material established by its creep low or its relaxation modulus. Given thevalue, the parameters of the material damage curve can be obtained from the stresses observed in the fatigue test. Hence, any deviation in output (stress or strain) from the theoretical value is re lected by Equation (6). Consequently, stress-pseudo-strain relationships provide a correspondence between linear viscoelastic undamaged and damaged bodies. All damage growth over time is depicted in a reduction in material stress.
The evolution law in Equation (7) re lects rate-effects through ! in $ ! and explicit timeeffects through the time-derivative of S on the left hand side. The only damage-related constitutive parameters to be determined for a given material are the function and the constant -. The validity of this assumption will be evaluated by a comparison of the algorithm with the experimental observation of response of the specimens.

Viscoelas2c Con2nuum Damage Theory
The continuum damage mechanics, originally developed for elastic materials, can be generalized for viscoelastic materials using the elastic-viscoelastic correspondence principle (Kim, 2009). The VECD model combines elements from the preceding sections to arrive at the constitutive relationship. From continuum damage, the stiffness reduction is de ined by the pseudo stiffness.
The WPT uses an internal state variable (S) to quantify damage. This internal state variable quanti ies any microstructural changes that result in the observed stiffness reduction.
For asphalt concrete under tensile stress, this damage variable is related primarily to the microcracking phenomenon.
The relationship between and is a material property that is independent of the mode of loading, temperature, and load amplitude and is referred as the damage characteristic curve (Kim, 2009). With these considerations, the nonlinear constitutive relationships are given by: The pseudo stiffness decreases from the value of 1 as the accumulated damage increases. According to Kim (2009), the expression for can be represented by a power law as: where and % are regression constants that it the curves obtained in the fatigue tests of the material. These constants characterize the material with respect to its susceptibility to damage.
Substituting / * into the Work Potential Equation by the derivative of Equation (9), and * by its incremental form, ∆* ∆ , one obtains: where, 8 and 8 + 1 are the steps at times 1 and 12 , respectively; 1 and 12 are damage parameter at 8 and 8 + 1, respectively; ∆ is the time interval between 1 and 12 . Equation (10) can be used to determine the grown of damage parameter in asphalt mixture. This parameter along with material pseudo stiffness will be used to obtain vs. curve, where: where 12 is the stress observed at time 12 and 12 ! is the pseudo strain at time 12 .

METHODOLOGY
Viscoelastic damage characterization refers to the determination of the characteristic damage relationship (i.e. the vs. curve). This paper presents a method for calculating the vs. curve obtained in direct tension tests considering sinusoidal controlled strain loading.

Sinusoidal Controlled Strain Rate Test
In this test, a controlled sinusoidal strain is applied while the stress response is measured. The strain loading is given by: where > is the angular frequency of the action and is the corresponding strain amplitude (peak to peak). The stress response of a linear viscoelastic material to this sinusoidal loading is given by (

Algorithm to Obtain Parameters L MM and L MN of the L vs O curve
The proposed algorithm is based on Equations (4), (9), (10), (11) and (13). It is assumed that the viscoelastic characterization of the asphalt mixture has been held previously, i.e., the values of the Prony series constants and the α value of the asphalt mixture are known. The algorithm consists of seven major steps presented in Figure 3.
Step 1 corresponds to the computation of the pseudo stiffness ( 12 ) for each measured stress ( 12 ), with the pseudo strain ( 12 ! ) evaluated dividing corresponding viscoelastic stress, computed using Equation (13), by ! according Equation (4). The initial damage parameter ( ) and the initial parameters and % chosen by the user, are prescribed in Steps 2 and 3, respectively. The damage parameter for each time step ( 12 ) is evaluated in Step 4 using the current values of parameters P and % P . In Step 4, a set of ordered pairs ( P , P for iteration R are itted to Equation (9) using the Least Squares Method, obtaining new values for the following iteration: P2 and % P2 . Then, the algorithm returns to Step 4 with the new values of and % , repeating the procedure. The iterations stop when the relative difference between the values of and % between two consecutive iterations is less than or equal to a chosen tolerance (STU). The tolerance adopted in this paper is 10 -5 .
This algorithm was implemented in a program (CalCxS) developed to obtain the damage curve from laboratory test results or computer simulations of fatigue tests. In this program, the value of the initial damage parameter ( ), constant with each attempt, is arbitrated by the user. Therefore, it is recommended to run the program a few times with different values of . It is up to the user to choose the most appropriate curve according to the value of V % obtained in each attempt. This aspect is not a major issue, as good results are obtained in a few attempts.

Adjustment (L = M − L MM O L MN ) curve with the parameters L MM and L MN
The parameters and % of the curve represented by Equation (9) that best it the set of ordered pairs on iteration R, ) 1 P , 1 P +, are calculated using the Least Squares Method. Equation (9) can be written as: Taking the natural logarithm on each side of Equation (14) yields: Equation (15) resembles the equation of a straight line, X = Y + Y Z, where: Finally, after obtaining the coef icients Y and Y of the least squares regression line, the new estimations of the damage curve parameters are computed as follows:

RESULTS
To verify the proposed algorithm, fatigue tests with controlled sinusoidal strain were simulated using an axisymmetric model for three mixtures whose Prony series and the parameters of the vs. curve are known (Souza, 2018). Among the results of these simulations are the material integrities on the peak of each periodic action cycle. These integrities ( ) with the characteristics of the strain fatigue test simulation (frequency, amplitude, duration) were used to obtain the referred parameters of the vs. curve, with the presented algorithm. The fatigue test was simulated using the AEDCISO program (Almeida, 2019), which uses the S-VECD theory (Underwood et al., 2009). The AEDCISO program can simulate the fatigue test considering the stiffness degradation during the evolution of material damage. However, in the simulations performed for this article, this effect was not considered in order not to change the characteristics of the vs. original curve. As this is not the purpose of this article, no further details on the methodology used in the AEDCISO program is presented. The parameters of the vs. curve were obtained with CalcCxS program (Almeida, 2019). The constants of the Prony series of the mixtures are shown in Table 1. The α values with the vs. curve parameters are presented in Table 2 (Souza, 2018). In all simulations of fatigue test it was used the initial damage parameters, = 100. In each mixture, the frequencies 0.1; 0.5; 1.0; 5.0; 10.0 and 25.0Hz were considered with duration of 50000, 10000, 5000, 1000, 500 and 200 seconds, respectively, producing 5000 sine deformation cycles with amplitude (peak to peak), = 200 x 10 ] . In the laboratory, the tests must be performed with small strains that do not produce immediate failure in the specimens. In the case of computer simulation, this is not necessary. Thus, it was opted for strain amplitude value that produced the damage in a relatively small number of cycles.
The relative tolerance of 10 ^ was considered in the algorithm stop criterion for obtaining parameters and % of the vs. curve. To demonstrate the ability of the algorithm to ind the parameters of the vs. curve, initial approaches to and % were arbitrated very different from their known values. It was adopted = % = 0.01, but other values (within the range 0 to 1) can be used. The values of and % depend on and α associated with the vs. curve, as well as the stress units used in the test or in the simulation. To compare the resulting curves with the original curves, it was used in CalcCxS program, the same α values and initial damage values ( ) close to those used in the fatigue tests simulations performed with the AEDCISO program. This is not always possible as they eventually produce and % results that generate negative V % values. Thus, other values of were adopted to produce V % values close to 1.0.

CONCLUSIONS
This paper presents an algorithm to determine the damage characteristic curve of asphalt mixtures. Results demonstrate the ef iciency of the algorithm proposed, since the parameters obtained for different frequencies produce approximately equal vs. curves. The algorithm converges to the damage curve parameters, even though the initial approximations of and % are different from their original values, with R 2 values close to 1.0. All values of the parameter used in the algorithm are lower than the value of used in the fatigue test simulations. It was tried, in the CalcCxS program, the use of values equal to the value used in the simulations of fatigue tests, but it produced over low during the running of the program. In this case, the program issues a message requesting another value. Even so, the curves obtained by the algorithm adequately represent the original curves. The calculation of the pseudo strains with the use of the LVE theoretical responses in stress for a strain controlled action has the advantage to facilitate subsequent treatment of the data to obtain the vs. curve.