# S-N Curve Models for Composite Materials Characterisation: An Evaluative Review

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Criteria for S-N Curve Model Evaluation

_{a}) vs mean stress (σ

_{mean}) diagram, the stress ratios are divided into four ranges depending on loading type: R = 0–1 under tension-tension (T-T) loading, R = 0–−1 under tension-compression (T-C) loading, R = −1–−∞ under compression-tension (C-T) loading, R = −∞–1 under compression-compression (C-C) loading. The adaptability can be evaluated for capability of curve fitting by varying values of fitting parameters in S-N curve model at different stress ratios.

_{f}) has existed confusingly in many different forms as listed by Hwang and Han [20], Fatemi and Yang [40], Yang and Fatemi [41], Bathias and Pineau [42], and Christian [43]. A clear difference, though, between damage at failure (D

_{f}) and damage prior to failure (D) should be made to clarify D

_{f}[27]. Damage at failure (D

_{f}) is directly related to an S-N curve which is a locus of failure points caused by damage. Thus, an S-N curve itself indicates damage at failure in a way. Other quantities such as residual strength and residual modulus [44] may be able to represent the damage as well. More importantly, however, if the ultimate purpose of damage D is to predict the remaining fatigue life when applied stress varies, a damage function (D), which is a function of N and applied peak stress σ

_{max}, is required to satisfy the compatibility condition [27] to ensure that σ

_{max}and N compatibly contribute to the damage. If an S-N curve is derived from fatigue damage and to predict the remaining fatigue life involving independent variables (i.e., N

_{f}and σ

_{max}), the damage at tensile failure D

_{fT}is required to satisfy

_{fC},

_{f}= 0 or 0.5 or 1 at σ

_{uT}in the absence of a rigorous rational basis while some earlier researchers have ignored the initial boundary condition. Eskandari and Kim [27] recently provided a rationale for R = 0 to be N

_{f}= 0.5 at σ

_{uT}. However, it has not generally been settled down on other stress ratios yet. Nonetheless, it is necessary that N

_{f}at σ

_{uT}should be 0 < N

_{f}< 1 or adjustable for the future development until a rational value is found while N

_{f}= 0 should not be used not only because it is physically impossible to be a failure point but also because it does not exist on log coordinate axis.

_{max}and N

_{f}without a numerical scheme.

## 3. S-N Curve Models

#### 3.1. For Fatigue Data Characterisation

_{max}= applied peak stress, α, β = model fitting parameter, and N

_{f}= number of cycle at failure.

_{max}versus logN

_{f}.

_{∞}) may not exist [53], Stromeyer [54] added an additional term, σ

_{∞}, to Equation (3):

_{max}= applied peak stress, α, β = model fitting parameter, and N

_{f}= number of cycle at failure.

_{max}= applied peak stress, σ

_{∞}= fatigue limit, α, β, γ = model fitting parameter, N

_{f}= number of cycle at failure, and δ = number of stress cycles to which the balls or the grooves are subjected during one revolution of the bearing.

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength, σ

_{∞}= fatigue limit, α, β = model fitting parameter, N

_{f}= number of cycle at failure, and ψ = Gaussian error function, and subsequently proposed an S-N curve model;

_{uT}) and the fatigue limit (σ

_{∞}). The two fitting parameters (α and β) in Equation (8) can be determined by taking log on both sides of,

_{f}= 1 at σ

_{max}= σ

_{uT}and is efficient for determining the fitting parameters although the fatigue damage was not considered. The fatigue limit in the equation, which may not exit, is a disadvantage for curve fitting. However, the assumed fatigue limit may be regarded as being different from a fitting parameter because they can easily be estimated for fitting purposes whereas fitting parameters can be found on a trial and error basis in some cases without much idea on the initial trial value unless a known algorithm is available.

_{max}= applied peak stress, α, β = model fitting parameter, and N

_{f}= number of cycle at failure, for an S-N curve to derive a “damage equation”, which is far from being accurate for data fitting although an initial boundary condition at the first load cycle can be managed to be met.

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength α, β = model fitting parameter, and N

_{f}= number of cycle at failure.

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength α, β = model fitting parameter, and N

_{f}= number of cycle at failure.

_{f}should be zero for σ

_{max}= σ

_{uT}, which is not possible on logarithmic coordinates used for N

_{f}.

_{f}= number of cycle at failure, and C = constant.

_{max}) instead of the maximum stress for an S-N relation,

_{f}) = probability of survival.

_{f}) for fatigue characterization, unless other aspects are dealt with (e.g., thermos-mechanical fatigue in [59]), because the strain needs to be converted into stress for an S-N curve.

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength, σ

_{∞}= fatigue limit, α, β, γ = model fitting parameters, and N

_{f}= number of cycle at failure.

_{∞}) regions while Equation (19) extends to the low-cycle region. The Kohout and Vechet model has generally three straight lines to describe the assumed S-N curve in log-log coordinates, including two asymptotes with zero slope for tensile strength and fatigue limit (σ

_{∞}), and a tangent for the region described by the Basquin model. The points of intersection of the tangent with the two asymptotes occur at N

_{f}= α and N

_{f}= γ in the case of Equations (17) and (18). Thus, the parameters α and γ are numbers of cycles at which the S-N curve bends at high and low stresses respectively, and β is the slope of the tangent between the two bends (α and γ) in log-log coordinates. Kohout and Vechet model, therefore, is limited to the predetermined S-N curve. If the bend at a high stress does not exist at some stress ratios, it would not fit experimental data. Also, Equation (19) does not satisfy the initial boundary condition given that N

_{f}should be zero to be σ

_{max}= σ

_{uT}.

_{fT}), which satisfies Equation (1):

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength, α, β = model fitting parameters, and N

_{f}= number of cycle at failure.

_{f}) in this equation can be obtained by integration, yielding an S-N curve as a function of applied peak stress (σ

_{max}):

_{0}in the equation is to adjust the initial number of cycles for the first cycle failure point according to the quantized analysis—it is not possible to find it with the continuity concept of S-N curve [27]. For example, N

_{0}= 0.5 cycle at σ

_{max}= σ

_{uT}with R = 0. Thus, the Kim and Zhang model satisfies the initial boundary condition. The fitting parameters for S-N curve in Equations (22) and (23) are identical to those for damage rate with respect to N

_{f}(Equation (21)), which is an advantage for understanding a relation between damage rate and S-N curve. The two parameters (α and β) in Equations (22) and (23) can be determined for a set of fatigue data, given that Equation (21) is numerically,

_{max(i)}) at failure, ΔD

_{f(i)}= D

_{f(i-1)}− D

_{f(i)}for D

_{f(i-1)}> D

_{f(i)}or |1 − σ

_{max(i-1)}/σ

_{uT}| > |1 − σ

_{max(i)}/σ

_{uT}|, ΔN

_{f(i)}= N

_{f(i-1)}− N

_{f}

_{(i)}for N

_{f(i-1)}> N

_{f(i)}, and σ

_{max(i)}= (σ

_{max(i)}− σ

_{max(i-1)})/2.

_{f(i)}/ΔN

_{f(i)}and the corresponding σ

_{max(i)}. One of the ways is to directly use experimental data points on an S-N plane by choosing a pair of close data points for each value. The other way is to draw the best fit line manually through S-N data points for digitization (http://arohatgi.info/WebPlotDigitizer/app/) and then collect a set of digitized points. Then, log(ΔD

_{f(i)}/ΔN

_{f(i)}) is plotted as a function of log σ

_{max(i)}such that β is the slope and 10

^{α}is an intercept. It is practically efficient with, if necessary, a minimal iteration for the best fit. Figure 2 shows the sequence for plotting the best fit S-N curve.

_{f}= 1 for predicting the remaining fatigue life under spectrum loading.

_{max}at χ.

_{f}as shown in Figure 3 with the values as used: α = 0.003, β = 1, γ = 8.5, and σ

_{B}= (σ

_{uT}+ |σ

_{uC}|)/2. Thus, the fitting capability of this model is low. It has three fitting parameters (α, β, and γ) and one conditional property parameter (σ

_{B}). In other words, if σ

_{uT}< |σ

_{uC}| then σ

_{B}= σ

_{uC}; and if σ

_{uT}> |σ

_{uC}| then σ

_{B}= σ

_{uT}. If a chosen value in this way does not allow a good curve fitting, an alternative value σ

_{B}between σ

_{uT}and |σ

_{uC}| is required. Therefore, the efficiency of curve fitting for modelling is poor because of the number of fitting parameters and conditions imposed. The authors do not suggest a method of how to determine fitting parameters. Kawai and Itoh in 2014 suggested a similar S-N curve expression [22]:

#### 3.2. As a Function of Stress Ratio

_{f}) for a carbon fiber reinforced laminate made of 8 plies [45/90/-45/0/0/-45/90/45] with an average tensile strength of 586 MPa:

_{max}= applied peak stress, and σ

_{uT}= ultimate tensile strength.

_{uT}= ultimate tensile strength, and N = Number of cycles before failure.

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength, γ = fitting parameter, N

_{f}= number of cycle at failure, and R = stress ratio.

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength, N

_{f}= number of cycle at failure, and R = stress ratio.

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength, α′, β = model fitting parameters, and N

_{f}= number of cycle at failure.

_{uT}/σ

_{max}− 1)/(1 − R) in Equation (32) was experimentally found to be constant for R = 0.1–0.7, allowing us to determine α and β ratios using data sets from different stress ratios. For a constant stress ratio, two fitting parameters need to be determined for an S-N curve. Caprino and Giorlea in 1999 [68] further experimentally verified this for R = 10 and 1.43. Equation (34) allows the initial boundary condition to be N

_{f}= 1 at σ

_{uT}and the fitting parameters have a relation with Equation (31) for damage.

_{max}= applied peak stress, σ

_{uT}= ultimate tensile strength, α, β = model fitting parameters, f = loading frequency, N

_{f}= number of cycle at failure, and R = stress ratio.

#### 3.3. Associated with Stress Intensity Factor

^{1/2}), and A and m = constants.

_{f}) and initial crack length (a

_{0}) assuming that geometry factor Y is constant:

_{f}and a

_{0}with a

_{f}= a

_{0}for the first cycle fracture and a

_{f}= t for a low stress fracture:

_{f}in Equation (38) for Equation (39). The initial crack length (a

_{0}) in Equation (38) was regarded as a constant independent of applied stress level. The initial crack length a

_{0}may not be conceptually applicable for continuous fiber reinforced composites with a multiple crack damage pattern. However, the initial crack length (a

_{0}) may be regarded as an equivalent initial crack to the damage consisting of multiple cracks, given that the S-N curve behavior pattern may be for both single and multiple crack damages for un-notched specimens. The Wyzgoski and Novak model may be useful for providing an insight into internal damage and understanding S-N curve in relation to the stress intensity factor. At the same time, it should be noted that the stress intensity factor is limited to the tensile stress, implying that applicability of this model is not certain for stress ratios involving compressive stress, and may not be practical for S-N curve fitting due to the number of fitting parameters if A and m are treated as the fitting parameters. Figure 4 shows experimental data for injection molded PBT reinforced with 30w% short glass fibers [73] (obtained at R = 0.1 with σ

_{uT}= 100 MPa, specimen thickness (t) = 0.0032 m) in comparison with an S-N curve plotted in dashed line using Equation (38) for: Paris equation constants A = 1.53 × 10

^{−10}and m = 7.41, geometry factor Y = 1.17 for a semi-circular surface flaw [74], initial flaw size (a

_{0}) = 0.354 mm. Also, another S-N curve plotted with Kim and Zhang model Equation (22) or (23) with α = 4.9 × 10

^{−35}and β = 15.028 is shown in solid line. The Wyzgoski and Novak model appear reasonable for curve fitting although it is not as accurate as the Kim and Zhang model (see below for capability).

## 4. Evaluation of S-N Curve Model Capability

_{max}will be replaced with |σ

_{min}|, and σ

_{uT}with |σ

_{uC}| for evaluation.

#### 4.1. S-N Curve Models for Data Characterisation

_{uT}= 829 MPa obtained from Weibull [18] with 111 rotating beam tests at R = −1 is plotted in Figure 5 with data points having a probability of 50% failure. Two curves from the Weibull model Equation (8) with and without fatigue limit (σ

_{∞}) for respective two sets of fitting values—α = 21 × 10

^{−4}and β = 3.8 with an assumed σ

_{∞}= 353 MPa (provided by Weibull), and α = 0.003 and β = 3.1 with an assumed σ

_{∞}= 0 (to see if there would be any difference)—are plotted in comparison with other S-N models to be discussed. (Please note that the plotted lines from some of models are overlapped and therefore not clearly distinguishable.) The capability of fitting with a fatigue limit (σ

_{∞}) appears to be very good but, with σ

_{∞}= 0, it is relatively poor at a high N

_{f}. The best fit with the fatigue limit by Weibull model may be regarded as the reference for comparisons with other S-N curves. Plots generated by other S-N models are shown with: α = 0.001 and β = 0.093 for Sendeckyj model Equations (11) and (12); α = 35 and β = 0.21 for Hwang and Han model Equations (13) and (14); α = 776.25 and β = −0.0895 for Kohout and Vechet model Equation (19); and α = 10

^{−38.44}and β = 11.809 for Kim and Zhang model Equations (22) and (23) with N

_{0}= 0.5. The fitting capabilities for models by Weibull, Sendeckyj, Kohout and Vechet, and Kim and Zhang appear to be practically identical to each other and may be regarded as being sufficiently good for practical data characterization. In contrast, the fitting capability of Basquin model Equation (3) is found to be poor as expected when σ

_{uT}is included in data despite the fact that it has still been used by many researchers (e.g., refs. [75,76,77] to name a few). The fitting capability of Hwang and Han model appears to be also poor because of its limited capability—as the α value increases in Equations (13) and (14), σ

_{max}approaches the ultimate strength at N

_{f}= 1 but other part of the curve was found to deviate increasingly from the experimental data. A reasonable fit of Hwang and Han model can only be achieved by omitting the data points near the ultimate strength. The Basquin, and Hwang and Han models were accordingly eliminated from further comparison with other models at other stress ratios.

_{uT}= 2013 MPa is given in Figure 6. Weibull model Equation (8) was plotted with α = 0.088 and β = 1.9 for an assumed fatigue limit of 250 MPa, Sendeckyj model Equations (11) and (12) with α = 0.0485 and β = 0.157, Kohout and Vechet model Equation (19) with α = 19.953 and β = −0.155, and Kim and Zhang model Equations (22) and (23) with α = 3.13 × 10

^{−27}and β = 7.381 for N

_{0}= 0.5. The S-N curve of Sendeckyj model may be regarded as the best fit as already demonstrated in [19]. The fitting capabilities of the four models still appear to be equally good and practically identical to each other.

_{uC}= 807 MPa and σ

_{uT}= 1887 MPa for T-C loading at R = −0.43. For comparison purposes of the models, the original experimental data point for N

_{f}= 0.5 at σ

_{uC}was changed to N

_{f}= 1 because of the sensitivity of the S-N curve models to the experimental data point for the initial N

_{f}at σ

_{uC}also because Sendeckyj and Weibull models are designed to take N

_{f}= 1 at σ

_{uC}although Kohout and Vechet model Equation (19) is to take N

_{f}= 0 at σ

_{uC}while Kim and Zhang model is capable of taking N

_{f}= 1 cycle at σ

_{uC}as well, (Note N

_{f}= 0 is not possible on logarithmic coordinate as discussed earlier.) Accordingly, N

_{0}in Kim and Zhang model Equations (22) and (23) was set to be 1. As a result, the model fitting parameters were found to be α = 2.5 × 10

^{−1}and β = 0.83 for Weibull model Equation (8) with an assumed σ

_{∞}= 0, α = 4.485 and β = 0.078 for Sendeckyj model Equations (11) and (12), α = 0.35and β = −0.08 for Kohout and Vechet model Equation (19), and α = 2.884 × 10

^{−160}and β = 54.388 for Kim and Zhang model Equation Equations (22) and (23). The fitting capabilities of the three models (i.e., Weibull, Sendeckyj, and Kim and Zhang) appear equally good and practically identical. However, as discussed earlier, Kohout and Vechet model Equation (19) is found to be not capable of satisfying the boundary condition at σ

_{uC}not because of the experimental data point adjusted to be N

_{f}= 1 at σ

_{uC}but because of the predetermined S-N curve shape as seen near the first cycle (at log N

_{f}= logα = log0.35 = −0.4559 for the bend) although the degree of deviation from other models is not significant in this case. If the fatigue data point more rapidly decreases at low cycles, the Kohout and Vechet S-N curve would have more tendency of deviating from the data for the same reason. Also, the points of intersection of the tangent with the two asymptotes at log N

_{f}= logα for the bend of S-N curve is arbitrary to some extent.

_{uC}= 807 MPa and σ

_{uT}= 1887 MPa in comparison with plots from the various S-N curve models. The S-N line shape tendency is similar to those in Figure 5 and Figure 6 such that the fitting capabilities of all the models are again equally good without much sensitivity for the initial N

_{f}at σ

_{uC}. In the case of Kohout and Vechet model Equation (19), when α is high for the bend of S-N curve, the sensitivity to the initial experimental point N

_{f}at σ

_{uC}decreases as the σ

_{min}substantially approaches the σ

_{uC}. The model fitting parameters without adjusting the initial experimental point N

_{f}= 0.5 at σ

_{uC}were found to be α =3.5 × 10

^{−3}and β = 3.59 for Weibull model Equation (7) with an assumed σ

_{∞}= 150 MPa, α = 0.001985 and β = 0.08 for Sendeckyj model Equations (11) and (12), α = 398 and β = −0.0775 for Kohout and Vechet model Equation (19), and α = 4.295 × 10

^{−38}and β = 13.506 with N

_{0}= 0.5 for Kim and Zhang model.

_{uC}= 807 MPa, (σ

_{uT}= 1887 MPa) is shown in Figure 9. For comparison purposes of the models, an experimental data point is adjusted again to be N

_{f}= 1at σ

_{uC}for the same reason in the case of R = −0.43 shown in Figure 7 for Weibull, and Secdeckyj models except Kohout and Vechet model Equation (19). Also, N

_{0}in Kim and Zhang model Equations (22) and (23) was set to be 1. The model fitting parameters were found to be α = 0.138 and β = 0.66 for Weibull model Equation (7) with an assumed σ

_{∞}= 185 MPa, α = 1.3 and β = 0.019 for Sendeckyj model Equations (11) and (12), α = 1and β = −0.02 for Kohout and Vechet model Equation (19), and α = 2.88 × 10

^{−160}and β = 54.388 for Kim and Zhang model with N

_{0}= 1. The fitting capabilities of all the models appear to be equally good and practically identical indicating that all the models can fit the approximate straight line. However, the inherent nature of Kohout and Vechet model from the predetermined S-N curve is again detectable due to the pre-determined points of intersection of the tangent with the two asymptotes at logN

_{f}= logα = log1 = 0 for the bend of S-N curve of the model.

#### 4.2. S-N Curve Models for Predicting Stress Ratio Effect

_{f}= 1 at σ

_{max}= σ

_{uT}while Poursatip and Beaumont model Equations (29) and (30) adopts N

_{f}= 0 at σ

_{uC,}which does not exist on logarithmic coordinate. Therefore, the same condition i.e., N

_{f}= 1 at σ

_{max}= σ

_{uT}as those in Figure 7 and Figure 9 was employed for comparison purposes such that the other models and experimental data were accordingly adjusted. Experimental data and plots will be in lin-log coordinates.

^{−39}and β = 11.809 for N

_{0}= 1; Poursatip and Beaumont model Equation (30) with α = 17,000 and an added N

_{0}= 1; D’Amore model Equation (34) with α′ = 0.053 and β = 0.2; and Epaarachchi and Clausen model Equation (36) with α′ = 0.0007 and β = 0.245. It is seen that all the models do not agree with the reference performed by the three models or Kim and Zhang model. The Poursatip and Beaumont model, however, appears closer to the reference fit than the other two models. It was found, at the same time, that the S-N line shape of Poursatip and Beaumont model is controlled by a single fitting parameter α which only allows it to move around the first data point (for N

_{f}= 1 at σ

_{max}= σ

_{uT}) as the hinge without much controlling the shape. On the other hand, the fitting capabilities of D’Amore, and Epaarachchi and Clausen models appear relatively poor, given that radii of the curvature of the S-N curves at low cycles are too large compared to the reference fit. A possible reason for this is that the derivation may not be validly conducted as discussed earlier. Also, it would not be surprising to find that D’Amore, and Epaarachchi and Clausen models are practically identical in this case probably due to their similarities in expression and derivation [see Equations (34) and (36)]. The only difference made by the exponent 1.6 in the Epaarachchi and Clausen model Equation (36) is found to be reflected only in values of fitting parameters α′ and β.

_{uT}= 2013 MPa is shown in Figure 11 in comparison with S-N curves generated from the various models for R = 0.1 under T-T loading. The fitting parameters were found as follows: α = 10

^{−26.505}and β = 7.3813 for Kim and Zhang model Equations (22) and (23) with N

_{0}= 1; α = 60 and an added N

_{0}= 1 for Poursatip and Beaumont model Equation (30); α′ = 0.033 and β = 0.44 for D’Amore model Equation (34); α′ = 0.00035 and β = 0.51 for Epaarachchi and Clausen model Equation (36). D’Amore, and Eparrachchi and Clausen models appear improved for fitting capabilities compared to those for R = −1 (Figure 10) probably because of the rapid decrease in experimental data at low cycles as expected from their S-N curve performances shown in Figure 10 although clear deviations from the best fit are noticed. However, the added exponent 1.6 in Equation (36) does not seem to greatly improve the D’Amore model Equation (34). A fitting capability of Poursatip and Beaumont model appear still good although a slight difference from the best fit is noticed.

_{uC}= 807 MPa and (σ

_{uT}= 1887 MPa) for R = −0.43 in comparison with various models: Kim and Zhang model Equations (22) and (23) with α = 8.18 × 10

^{−41}and β = 13.6 for N

_{0}= 1; Poursatip and Beaumont model Equation (30) with α = 29 and added N

_{0}= 1; D’Amore model Equation (34) with α′ = 0.3 and β = 0.19; and Epaarachchi and Clausen model Equation (36) with α′ = 0.0169 and β = 0.14. The Epaarachchi and Clausen model appear very close to the reference fit (or Kim and Zhang model) while Poursatip and Beaumont, and D’Amore models noticeably deviated from the reference fit. The difference in behavior between D’Amore model, and Epaarachchi and Clausen model indicates that the added exponent 1.6 in Equation (36) noticeably affects the S-N curve shape in this case.

_{uC}= 807 MPa, (σ

_{uT}= 1887 MPa) for R = −3 representing C-T loading is shown in Figure 13 with S-N curves generated from the various models: Kim and Zhang model with α = 4.3 × 10

^{−38}, β = 13.506, and N

_{0}= 1; Poursatip and Beaumont model Equation (30) with α = 11,000 and added N

_{0}= 1; for D’Amore model Equation (34) with α′ = 0.053 and β = 0.2; for the Epaarachchi and Clausen model Equation (36) with α′ = 0.001 and β = 0.25. The fitting capabilities of all models appear to be good but differences between models are not as obvious as those for R = −1 in Figure 10 because the experimental data set is relatively small compared to that for R = −1. However, the same tendencies of D’Amore, and Epaarachchi and Clausen models are seen such that the radii of the curvature of S-N curve at low cycles is still large. Also, it is noted that D’Amore, and Epaarachchi and Clausen models behave similarly again without much effect of the modified exponent 1.6 in Equation (36) on the S-N curve shape.

_{uC}= 807 MPa (σ

_{uT}= 1887 MPa) for R = 10 with S-N curves generated from the various models: the Kim and Zhang model Equations (22) and (23) with α = 2.9 × 10

^{−160}and β = 54.388 for N

_{0}= 1; the Poursatip and Beaumont model Equation (30) with α = 719 and added N

_{0}= 1; the D’Amore model, Equation (34), with α′ = 0.043 and β = 0.01; the Epaarachchi and Clausen model Equation (36) with α′ = 0.0075 and β = 0.048. The fitting capabilities of the D’Amore, and Epaarachchi and Clausen models appear to be practically identical to the reference fit (or Kim and Zhang model). In contrast, the Poursatip and Beaumont model significantly deviates from the reference fit and seems to have a strong tendency of producing the concave-down S-N curve at low cycles. Judging from the earlier observations, the Poursatip and Beaumont model seems incapable of producing other shapes than the concave-down S-N curve at low cycles regardless of stress ratio. A possible reason for this may be due to some defects during derivation as discussed earlier even though Equation (27) satisfies Equation (1).

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Stress ratio ranges on σ

_{a}–σ

_{mean}diagram for the loading types: T-T, T-C, C-T, and C-C.

**Figure 3.**Experimental data from Kawai and Koizumi [21] for carbon fiber reinforced plastics with tensile strength = 782 MPa and compressive strength = 532 MPa obtained at R = −0.68. An S-N curve was fitted according to Equation (25) using α = 0.003, β = 1, γ = 8.5, and σ

_{B}= (σ

_{uT}+ |σ

_{uC}|)/2.

**Figure 4.**Wyzgoski and Novak [71] experimental data and S-N curve according to Equation (38) with t = 0.0032 m, m = 7.41, A =1.53 × 10

^{−10}m/cycle, Y = 1.17, R = 0.1, and a

_{0}= 0.354 mm. Another S-N curve was plotted according to the Kim and Zhang model Equation (23) with α = 1.53 × 10

^{−10}and α = 15.028.

**Figure 5.**Weibull experimental data for R = −1 plotted with data points having a probability of 50% failure for Bofors FR 76 Steel rotating beam test with σ

_{uT}= 829 MPa [18]. S-N curves were obtained from Weibull model with α = 21 × 10

^{−4}and β = 3.8 for an assumed σ

_{∞}= 353 MPa; Weibull model with α = 0.003 and β = 3.1 for an assumed σ

_{∞}=0; Sendeckyj model with α = 0.001 and β = 0.093; Hwang and Han model with α = 35 and β = 0.21; Kohout and Vechet model with α = 776.25 and β = −0.0895; and Kim and Zhang model with α = 10

^{−38.44}, β = 11.809 and N

_{0}= 0.5.

**Figure 6.**Sendeckyj experimental data at R = 0.1 [19] for T-T loading with S-N curves obtained from: Weibull model with α = 0.088 and β = 1.9 for assumed σ

_{∞}= 250 MPa; Sendeckyj model with α = 0.0485 and β = 0.157; Kohout and Vechet model with α = 19.953 and β = −0.155; and Kim and Zhang model with α = 3.13 × 10

^{−27}, β = 7.381 for N

_{0}= 0.5.

**Figure 7.**Kawai and Itoh experiment [22] with σ

_{uC}= 807 MPa and σ

_{uT}= 1887 MPa with S-N curves obtained from: Weibull model with α = 2.5 × 10

^{−1}and β = 0.83, and assumed σ

_{∞}= 0; Sendeckyj model with α = 4.485 and β = 0.078; Kohout and Vechet model with α = 0.35and β = −0.08; and Kim and Zhang model with α = 2.884 × 10

^{−160}, β = 54.388 and N

_{0}= 1.

**Figure 8.**Kawai and Itoh experimental data [22] for carbon fiber reinforced epoxy laminates with σ

_{uC}= 807 MPa and σ

_{uT}= 1887 MPa, and S-N curves obtained from: Weibull model with α = 3.5 × 10

^{−3}, β = 3.59, and assumed σ

_{∞}= 150 MPa; Sendeckyj model with α = 0.001985 and β = 0.08; Kohout and Vechet model with α = 398.1and β = −0.0775; and Kim and Zhang model with α = 4.295 × 10

^{−38}, β = 13.506, and N

_{0}= 0.5.

**Figure 9.**Kawai and Itoh experimental data [22] for carbon fiber reinforced epoxy laminates with σ

_{uC}= 807 MPa and σ

_{uT}= 1887 MPa, and S-N curves obtained from: Weibull model with α = 0.138, β = 0.66, and assumed σ

_{∞}= 185 MPa; Sendeckyj model with α = 1.3 and β = 0.019; Kohout and Vechet model with α = 1 and β = −0.02; and Kim and Zhang model with α = 2.88 × 10

^{−160}, β = 54.388 and N

_{0}= 1.

**Figure 10.**Weibull experimental data at R = −1 [18] plotted with data points having a probability of 50% failure for Bofors FR 76 Steel rotating beam test with σ

_{uT}= 829 MPa, and S-N curves obtained from: Kim and Zhang model with α = 3.63 × 10

^{−39}, β = 11.809, and N

_{0}= 1; Poursatip and Beaumont model with α = 17,000 and added N

_{0}= 1; D’Amore model with α′ = 0.053 and β = 0.2; and Epaarachchi and Clausen model with α′ = 0.0007 and β = 0.245.

**Figure 11.**Sendeckyj experimental data at R = 0.1 for glass reinforced epoxy laminates with σ

_{uT}= 2013 MPa [19] under T-T loading, and S-N curves obtained from: Kim and Zhang model with α = 10

^{−26.505}, β = 7.3813, and N

_{0}= 1; Poursatip and Beaumont model with α = 60 and added N

_{0}= 1; D’Amore with α′ = 0.033 and β = 0.44; and Epaarachchi and Clausen model with α′ = 0.00035 and β = 0.51.

**Figure 12.**Kawai and Itoh experiment at R = −0.43 [22] with σ

_{uC}= 807 MPa and σ

_{uT}= 1887 MPa, and S-N curves obtained from: Kim and Zhang model with α = 8.18 × 10

^{−41}, β = 13.6, and N

_{0}= 1; Poursatip and Beaumont model α = 29 and added N

_{0}= 1; D’Amore model with α′ = 0.3 and β = 0.19; and Epaarachchi and Clausen model with α′ = 0.0169 and β = 0.14. An experimental data point for N

_{f}= 0.5cycle was adjusted to have the initial condition N

_{f}= 1 at σ

_{uC}for comparison purposes.

**Figure 13.**Kawai and Itoh experimental data at R = −3 [22] for carbon fiber reinforced epoxy laminates with σ

_{Uc}= 807 MPa and σ

_{uT}= 1887 MPa, and S-N curves obtained from: Kim and Zhang model with α = 4.3 × 10

^{−38}, β = 13.506, and N

_{0}= 1; Poursatip and Beaumont model with α = 11,000 and added N

_{0}= 1; D’Amore model with α′ = 0.053 and β = 0.2; and Epaarachchi and Clausen model with α′ = 0.001 and β = 0.25.

**Figure 14.**Kawai and Itoh experimental data at R = 10 [22] for carbon fiber reinforced epoxy laminates with σ

_{uC}= 807 MPa and σ

_{uT}= 1887 MPa, and S-N curves obtained from: Kim and Zhang model with α = 2.9 × 10

^{−160}, β = 54.388, and N

_{0}= 1; Poursatip with α = 719 and added N

_{0}= 1; D’Amore model with α′ = 0.043 and β = 0.01; and Epaarachchi and Clausen model with α′ = 0.0075 and β = 0.048.

S-N Curve Models | Capability of Curve Fitting | Adaptability to Different Stress Ratios | Relation with Physical Properties | Remarks | ||
---|---|---|---|---|---|---|

Boundary at Initial N_{f} | Damage Representation | Satisfaction of $\frac{\partial {\mathit{D}}_{\mathbf{fT}}}{\partial {\mathit{\sigma}}_{\mathbf{max}}}=-\mathit{A}$ | ||||

Basquin (1910) | Poor | Poor | N/A | N/A | N/A | |

Stromeyer (1914) | Poor | Poor | N/A | N/A | N/A | |

Weibull (1952) | Good | Good | Yes | N/A | N/A | Fatigue limit is required |

Henry (1955) | Poor | N/A | N/A | N/A | N/A | |

Marin (1962) | Poor | N/A | N/A | N/A | N/A | Same form as Basquin model. |

Sendeckyj (1981) | Good | Good | Yes | N/A | N/A | |

Hwang and Han (1986) | Poor | Poor | No | N/A | N/A | |

Kohout and Vechet (2001) Equation (19) | Good | Good | No | N/A | N/A | With limited curve shaping (e.g., R = −0.43) |

Kim and Zhang (2001) | Good | Good | Yes | Yes | Yes |

**Table 2.**Summary of S-N curve models for data characterization proposed by Weibull, Sendeckyj, Kohout and Vechet, and Kim and Zhang.

Experimental Data | Loading | S-N Curve Models | Fitting Parameters | |||
---|---|---|---|---|---|---|

α | β | σ_{∞} (MPa) | N_{0} | |||

Weibull (1952) [18] with σ_{uT} = 829 MPa | R = −1 for T-C or C-T loading | Weibull (1952) ${\sigma}_{\mathrm{max}}=\left({\sigma}_{\mathrm{uT}}-{\sigma}_{\infty}\right)\mathrm{exp}\left[-\alpha {(\mathrm{log}{N}_{\mathrm{f}})}^{\beta}\right]+{\sigma}_{\infty}$ | 2.1 × 10^{−5} | 3.8 | 353 | - |

0.003 | 3.1 | 0 | - | |||

Sendeckyj (1981) ${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{uT}}/{(1-\alpha +\alpha {N}_{\mathrm{f}})}^{\beta}$ | 0.001 | 0.093 | - | - | ||

Kohout and Vechet (2001) ${\sigma}_{\mathrm{max}}={\sigma}_{\mathrm{uT}}{\left[({N}_{\mathrm{f}}+\alpha )/\alpha \right]}^{\beta}$ | 776.25 | −0.090 | - | - | ||

Kim and Zhang (2001) ${N}_{\mathrm{f}}={\sigma}_{\mathrm{uT}}^{-\beta}/[\alpha (\beta -1)]\left[{\left({\sigma}_{\mathrm{max}}/{\sigma}_{\mathrm{uT}}\right)}^{1-\beta}-1\right]+{N}_{0}$ | 10^{−38.44} | 11.81 | - | 0.5 | ||

Sendeckyj (1981) [19] σ_{uT} = 2013 MPa | R = 0.1 for T-T loading | Weibull (1952) | 0.088 | 1.9 | 250 | - |

Sendeckyj (1981) | 0.0485 | 0.157 | - | - | ||

Kohout and Vechet (2001) | 19.953 | −0.155 | - | |||

Kim and Zhang (2001) | 3.13 × 10^{−27} | 7.381 | - | 0.5 | ||

Kawai and Itoh (2014) [22] with σ_{uC} = 807 and σ_{uT} = 1887 MPa | R = −0.43 for T-C loading | Weibull | 2.5 × 10^{−1} | 0.83 | 0 | 1 |

Sendeckyj (1981) | 4.485 | 0.078 | - | 1 | ||

Kohout and Vechet (2001) | 0.35 | −0.08 | - | - | ||

Kim and Zhang (2001) | 2.88 × 10^{−160} | 54.39 | - | 1 | ||

Kawai and Itoh (2014) [22] with σ_{uC} = 807MPa and σ_{uT} = 1887 MPa | R = −3 for C-T loading | Weibull (1952) | 3.5 × 10^{−3} | 3.59 | 150 | - |

Sendeckyj (1981) | 1.985 × 10^{−3} | 0.08 | - | - | ||

Kohout and Vechet (2001) | 398.1 | −7.75 × 10^{−2} | - | - | ||

Kim and Zhang (2001) | 4.295 × 10^{−38} | 13.506 | - | 0.5 | ||

Kawai and Itoh (2014) [22] with σ_{uC} = 807 MPa and σ_{uT} = 1887 MPa | R = 10 for C-C loading | Weibull (1952) | 0.138 | 0.66 | 185 | 1 |

Sendeckyj (1981) | 1.3 | 0.019 | - | 1 | ||

Kohout and Vechet (2001) | 1 | −0.02 | - | - | ||

Kim and Zhang (2001) | 2.88 × 10^{−160} | 54.388 | - | 1 |

S-N Models | Google Scholar Citations | Citation Rates/Year |
---|---|---|

Basquin (1910) | 1311 | 12 |

Weibull (1952) | 14 | 0.21 |

Sendeckyj (1981) | 111 | 3.0 |

Kohout and Vechet (2001) | 73 | 4.3 |

Kim and Zhang (2001) | 15 | 0.88 |

S-N Curve Models | Capability of Curve Fitting and Applicability to Different Stress Ratios | Number of Fitting Parameters | Relation with Physical Properties | |||
---|---|---|---|---|---|---|

Boundary at Initial N_{f} | Damage Representation | Satisfaction of $\frac{\partial {\mathit{D}}_{\mathbf{fT}}}{\partial {\mathit{\sigma}}_{\mathbf{max}}}=-\mathit{A}$ | ||||

R | Capability | |||||

Poursartip and Beaumont (1986) ${N}_{\mathrm{f}}=\alpha {\left(\frac{{\sigma}_{\mathrm{max}}}{{\sigma}_{\mathrm{uT}}}\right)}^{-6.393}\left(\frac{{\sigma}_{\mathrm{uT}}-{\sigma}_{\mathrm{max}}}{{\sigma}_{\mathrm{uT}}}\right)$ | 1 | Good | 1 | Yes | Partially Yes | Yes |

0.1 | Good | |||||

−0.43 | Poor | |||||

−3 | Reasonable | |||||

10 | Poor | |||||

D’Amore et al. (1996) ${N}_{\mathrm{f}}={\left[\frac{{\sigma}_{\mathrm{uT}}-{\sigma}_{\mathrm{max}}}{{\alpha}^{\prime}{\sigma}_{\mathrm{max}}}+1\right]}^{1/\beta}$ | −1 | Poor | 2 | Yes | N/A | N/A |

0.1 | Reasonable | |||||

−0.43 | Poor | |||||

−3 | Reasonable | |||||

10 | Good | |||||

Epaarachchi and Clausen (2003) ${N}_{\mathrm{f}}={\left(\frac{{\sigma}_{\mathrm{uT}}-{\sigma}_{\mathrm{max}}}{{\alpha}^{\prime}{\left({\sigma}_{\mathrm{max}}\right)}^{1.6}}+1\right)}^{1/\beta}$ | −1 | Poor | 2 | Yes | N/A | N/A |

0.1 | Reasonable | |||||

−0.43 | Good | |||||

−3 | Reasonable | |||||

10 | Good |

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Burhan, I.; Kim, H.S.
S-N Curve Models for Composite Materials Characterisation: An Evaluative Review. *J. Compos. Sci.* **2018**, *2*, 38.
https://doi.org/10.3390/jcs2030038

**AMA Style**

Burhan I, Kim HS.
S-N Curve Models for Composite Materials Characterisation: An Evaluative Review. *Journal of Composites Science*. 2018; 2(3):38.
https://doi.org/10.3390/jcs2030038

**Chicago/Turabian Style**

Burhan, Ibrahim, and Ho Sung Kim.
2018. "S-N Curve Models for Composite Materials Characterisation: An Evaluative Review" *Journal of Composites Science* 2, no. 3: 38.
https://doi.org/10.3390/jcs2030038