Description of fatigue sensitivity curves and transition to critical states of polymer composites by cumulative distribution functions

A BSTRACT . In this paper


INTRODUCTION
redicting the remaining life of composite materials used in structures that can operate under cyclic loading represents a relevant scientific mission of deformable solid mechanics. A number of papers dedicates to experimental studies of mechanical characteristics of various classes of composites exposed to cyclic effects [1][2][3][4][5]. Staging of damage accumulation processes is noted in many of them [6][7][8][9]. The initial stage, also called the initiation stage, suggests fast damage accumulation. Multiple fatigue damages related to matrix cracking, damage to phase interfaces, and rupture of some fibers were found in [10][11][12][13]. The second stage, also called the stabilization stage, suggests slow damage accumulation. Some authors suggest that it involves matrix cracking processes. The stabilization stage is followed by intensified accumulation of damages and transition to the third stage, or the final breakdown stage, which implies fiber destruction and macro-failure of the specimen. Damage accumulation in the composites frequently leads to mechanical properties (Young's modulus, ultimate strength, etc.) reduction [14][15][16][17][18]. P A number of models which enable damage value calculations under fatigue loading exist. The most famous of them are the Palmgren-Miner model (linear summing of damages) [19][20][21][22] and the Marco-Starkie model (non-linear summing) [23][24][25][26][27]. However, neither of these models takes into account the aforementioned three stages, so formally they are not suitable to describe damage accumulation processes in composites. An approach where the damage parameter is related to the varying mechanical characteristic takes place in [5,9,[28][29]. Moreover, some papers [8,[30][31][32] model the degradation of composite mechanical characteristics by setting random values of strength and deformation parameters of the reinforcing component and matrix. The modeling results are good, but their disadvantage is the high number of required constants. This paper proposes a new model to describe the degradation of composite mechanical characteristics after preliminary cyclic loading. The model is based on experimental data approximation by probability distribution functions.

MATERIAL AND METHODS
wenty-six fiber-glass laminate specimens with the reinforcement pattern of [0/90] 8 were used in the experimental studies. The test method is based on the existing standards of quasi-static and fatigue tension of polymer composite materials. Nominal values of ultimate strength σ u and elasticity modulus E were taken from quasi-static uniaxial tension tests (ASTM D3039). The maximum number of cycles to failure N max was found for uniaxial cyclic tension at the maximum stress value σ max = 0.5·σ u , the asymmetry coefficient R = 0.1, and the frequency ν = 20 Hz (ASTM D3479). Three specimens were tested for quasi-static and cyclic tension. The other 20 specimens were exposed to preliminary cyclic loading and then statically tested. Preliminary cyclic exposure was implemented within 0.1 to 0.8 nominal fatigue life N max . The test method is schematically shown in Fig. 1. Figure 1: The order of experimental tests procedure: 1 -quasi-static tension tests; 2 -fatigue tests; 3 -preliminary cyclic loading; 4 -quasi-static tension after preliminary cyclic loading.
For each specimen, the fatigue sensitivity coefficients are found using the formula where E is Young's modulus; E 0 is the mean Young's modulus for a non-damaged material; σ u is the ultimate strength of the material; σ u0 is the ultimate strength of a non-damaged material. The fatigue sensitivity coefficient takes values from 0 (completely failed material) to 1 (non-damaged material). Fatigue sensitivity coefficients correspond to the following damages values The preliminary cyclic exposure is found using the formula  0 N n N (3) T where N is the number of preloading cycles; N 0 is the fatigue life for this loading cycle. The test data set represents the dependency of the fatigue sensitivity coefficient K B (K E ) on the preliminary cyclic exposure n. The results are processed using the model below.

MODEL DESCRIPTION
ypical points of the fatigue sensitivity curve in the coordinates of preliminary cyclic exposure vs. fatigue sensitivity coefficient (n -K B ) are shown in Fig. 2a. The following conversion can be made: the same points are built in the coordinates of damage value vs. preliminary cyclic exposure (ω B -n) as shown in Fig. 2b. Some features of this dependency can be noted. First, it is limited by zero and one. Second, if the "healing" of the material is absent, this dependency is monotone-increasing. Third, the characteristic segment of slow damage accumulation in the diagram middle can be noted. These features also have some probability distribution integral functions. In the case of non-damaged material before preliminary cyclic exposure, the n(ω B ) function passes through the coordinate system center. Therefore, consideration of the two-parameter Weibull law of probability distribution [33] and beta distribution is convenient. As an example, the integral curve of the Weibull distribution law is given in Fig. 2c.

Two-parameter Weibull distribution
The dependency of the preliminary cyclic exposure on the damage can be described by the following equation: where λ>0 is the scale parameter; κ>0 is the form parameter. Both these values are material properties characterizing its ability to keep strength and rigidity after some operating time. In a general case, these parameters can depend on the temperature, exposure amplitude, frequency, etc. The dependency of the residual strength coefficient on the preliminary cyclic exposure can be described as: The parameters λ and κ can be defined both numerically and using the method of least squares from the equation of a straight line approximating data in logarithmic coordinates (these approaches give close but slightly different results): x T An example of this dependence is shown in Fig. 3. To divide the fatigue sensitivity curve into damage accumulation stages, the derivative of the damage value function can be considered: The graph of this function is given in Fig. 4a. The physical sense of this function is the damage accumulation rate. For a low number of cyclic exposures, the intensity of damage accumulation is high (stage I), this stage is followed by an area of slow damage accumulation (stage II); when the number of cyclic exposures approaches the limit, the damage accumulation rate rapidly grows (stage III). Earlier, the authors in [9] proposed a definition of boundaries for these stages using the points n s1 and n s2 , where ω B '=0.3. Such division is conditional and may vary depending on the material (and its class). The values of n s1 and n s2 are defined by solving the transcendent Eqn. (7). An example of a K B (n) curve with the highlighted stages is given in Fig. 4b. The values of the fatigue sensitivity coefficient in exposures n s1 and n s2 are designated as K Bs1 and K Bs2 , respectively. The characteristic of the material defining the average rate of fatigue sensitivity coefficient reduction in the area of slow damage accumulation ψ can be introduced as: The model has some disadvantages. First, when n→1, K B becomes less than 0. However, this will take place only for the number of cycles close to the limit, so this specific feature can be neglected. Second, taking the logarithm is not suitable to describe data where the values of K B and n exceed 1. This problem can be solved by a numerical search of the parameters from Eqn. (5), where taking the logarithm is not required.

Beta distribution
The dependency of K B (n) can be described as follows where B n (α,β) is an incomplete beta-function; B(α,β) is a beta-function. The parameters α and β must exceed zero and are numerically searched. A derivative of the damage function: In comparison with the previous one, this function has some advantages. First, for n=1, the value K B =1. Second, all experimental points can be used to find the parameters α and β without the logarithm taking.

RESULTS AND DISCUSSION
he experimental data are given in Tab. 1. The average Young's modulus of a non-damaged material E 0 =2.29 GPa, the average ultimate strength of a non-damaged material σ u =381 MPa, and the average durability for this amplitude N 0 =213448 cycles. Approximation function parameters were found for the following cases: Weibull distribution, using data approximation in logarithmic coordinates (WL); Weibull distribution, the numerical search of parameters (WN); beta distribution, the numerical search of parameters (BN). For all cases, the determination coefficient was found. Tab. 2 and 3 contain the model parameters found for the full data set. Because R 2 >0.7 for all cases, the model has a high descriptive capability. Close determination coefficient values for all the cases can be noted. Fig. 5 represents curves of fatigue sensitivity coefficient and damage accumulation rate for Young's modulus (a, c) and ultimate strength (b, d) decrease. The similarity of these curves can be noted. Rationality and expediency of using probability distribution functions as the approximation of experimental data on mechanical characteristics reduction after preliminary cyclic exposure is concluded. A significant advantage of this approach is the small amount of required experimental data for parameter definition (4 tests required) and simplicity in modeling.

CONCLUSIONS
 An ability to describe experimental data on deformation and strength properties reduction of composites after preliminary cyclic exposure using probability distribution functions is considered.  Two probability distributions are considered: two-parameter Weibull law and beta distribution. For the Weibull distribution, a method to find parameters by approximation of experimental data in logarithmic coordinates is described.  Using of damage value function derivative is proposed to find damage accumulation stages boundaries.  Experimental data are processed for reduced mechanical characteristics of structural fiber-glass laminate. Model parameters are obtained. Fatigue sensitivity coefficient and damage accumulation rate curves are built. Boundaries of damage accumulation stages are defined. Determination coefficients are calculated. Because R 2 exceeds 0.7, the high descriptive capability of the model can be noted.  All parameter calculation methods showed close results. Beta distribution usage is more perspective due to the function having a value from 0 to 1. Each model requires only 4 experiment tests to define parameters (1 static test, 1 fatigue test, and 2 intermediate points on the curve). These models can be used in the modeling of the deformation and failure processes of various composite structures.  The future planned research will contain this method's usage for the description of other experimental data and loading conditions. Moreover, experimental study of the fast reduction of properties stages seems expedient for the purpose of more competent description.