Front. Mech. Eng. 2015, 10(3): 255–262 DOI 10.1007/s11465-015-0342-1
RESEARCH ARTICLE
Jorge Alberto Rodriguez DURAN, Ronney Mancebo BOLOY, Rafael Raider LEONI
Some remarks on the engineering application of the fatigue crack growth approach under nonzero mean loads
© Higher Education Press and Springer-Verlag Berlin Heidelberg 2015
Abstract The well-known fatigue crack growth (FCG) curves are two-parameter dependents. The range of the stress intensity factor ΔK and the load ratio R are the parameters normally used for describing these curves. For engineering purposes, the mathematical representation of these curves should be integrated between the initial and final crack sizes in order to obtain the safety factors for stresses and life. First of all, it is necessary to reduce the dependence of the FCG curves to only one parameter. ΔK is almost always selected and, in these conditions, considered as the crack driving force. Using experimental data from literature, the present paper shows how to perform multiple regression analyses using the traditional Walker approach and the more recent unified approach. The correlations so obtained are graphically analyzed in three dimensions. Numerical examples of crack growth analysis for cracks growing under nominal stresses of constant amplitude in smooth and notched geometries are performed, assuming an identical material component as that of the available experimental data. The resulting curves of crack size versus number of cycles (a vs. N) are then compared. The two models give approximately the same (a vs. N) curves in both geometries. Differences between the behaviors of the (a vs. N) curves in smooth and notched geometries are highlighted, and the reasons for these particular behaviors are discussed.
Received January 25, 2015; accepted June 15, 2015
✉
Jorge Alberto Rodriguez DURAN ( ) Mechanical Engineering Department, Federal Fluminense University, Volta Redonda 27255-125, Brazil E-mail:
[email protected] Ronney Mancebo BOLOY Mechanical Engineering Department, Federal Center of Technological Education, Angra dos Reis 23953-030, Brazil Rafael Raider LEONI Mechanical Engineer, Structural Data Acquisition & Fatigue Analysis Engineer, Resende 27511-970, Brazil
Keywords fatigue crack propagation modeling, life prediction, mean stress effects
1
Introduction
Brittle fracture in engineering components occurs when the stress intensity factor (SIF, variable K) of a cracked body is equal to the fracture toughness KIC of its composing material. This is a well-established procedure based on the linear elastic fracture mechanics (LEFM). Critically sized cracks rarely exist in new components. Most frequently, small cracks grow by subcritical mechanisms, including fatigue or environment-assisted cracking. The crack detection techniques currently available have a physical lower bound limitation for detecting and defining crack sizes. Classical design philosophies for fatigue assessment are based on elastic or elastic-plastic stress and strain analysis and comparison with empirical fatigue strength curves. In spite of the fact that fatigue tests are predominantly conducted until the specimen breaks apart, which implies in Nf= Ni+ Nif, these procedures do not account explicitly for the presence of cracks. For steel components under cyclic loads, however, cracks with sizes in the range of 7 to 700 μm might theoretically demand an LEFM procedure for their evaluation [1]. There are basically two situations in which crack analysis is necessary in the design phase: 1) Small cracks might already exist and relevant examples include welded structures or laminated products; 2) small cracks can nucleate early in the fatigue life, mainly in regions of high stress concentration. In both cases, an LEFM-based procedure should give better predictions because Nif Ni. Obviously, LEFM is not able to deal with crack sizes of the order of micrometers, mainly for cracks growing from notches [2], but some extensions of K, notablely the notch stress intensity factor (NSIF), are. Indeed, the mode I NSIF range has reduced the scatter band of fatigue strength data for welded joints when being compared with the nominal
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stress range [3]. Also, if the SIF range of small cracks that nucleate and propagate from the weld toe (a common fatigue failure mode for welded joints) is related to the NSIF parameter, the residual life calculated from the integration of Paris’ law can be used to unify the S vs. Nf (SN) fatigue data of many different weld geometries [4]. On the other hand, many safety relevant components or structures are designed to last for long periods of time. The aim of fatigue crack growth (FCG) analysis in the structural assessment phase is to determine safety factors in stress or life, wherein the crack size is considered to be the independent variable. As a result, properly inspection intervals can be defined. The underlying damage tolerant philosophy is largely based on the curves of crack size a vs. number of cycles Nf, which can be analytically obtained by direct or numerical integration of the crack propagation “law”, or measured experimentally. The needed SIFsolution is currently defined by numerical methods. Among many reports on the practical use of this philosophy, it is possible to mention the studies of Witek [5] and Luke et al. [6] that showed examples of application in compressor blades of the helicopter engine and in railway axles, respectively. To analyze FCG during design and structural assessment phases, FCG resistance curves are used. A general relation for these curves is da/dN = f(ΔK, R). The number of cycles Nif between the initial and final crack sizes ai and af, respectively, can be calculated by simple integration of this differential relation, or Nf
af
Ni
ai
da : ! dN ¼ Nf – Ni ¼ Nif ¼ ! f ðΔK,RÞ
(1)
The Paris’ relation, da/dN = CΔKmp, where C and mp are material parameters, is an approximation that fits the empirical data well in a particular region of the FCG resistance curve for a given load ratio. The Paris’ relation can be easily introduced into Eq. (1) and used for designing purposes. For other load ratios, however, the fitting constants (C and mp) should not be the same. Indeed, it is not possible to obtain empirical data for the whole loadratio range. Much effort has been expended to reduce the dependence of Eq. (1) to just one variable, such as replacing ΔK with another FCG driving force that includes the load-ratio effect. The dependence can also be reduced by using a set of fitting constants for all load ratios; in Paris’ form, this means obtaining a C(R) or ΔK(R) function. Among the well-known methods for dealing with the load-ratio effects on FCG curves are the Walker approach [7] and the more recent unified approach [8,9], which are briefly described in the following sections. The aim of the present paper is to evaluate the Walker and the unified approaches, while giving numerical examples of the complete process for calculating the
residual life of any cracked engineering component loaded in fatigue at any load ratio. First, for each approach, the steps for obtaining the material properties or calibration constants from a series of experimental FCG rate versus ΔK data at different mean loads are described for a 2024 T3 aluminum alloy. Then, two simulations of FCG under a constant range of stress and a fixed R-ratio (R = 0.4) are made, one in a smooth plate with an edge crack and the other one in a wide plate with a circular hole with a pair of cracks emanating from the hole. The number of cycles needed to propagate these cracks between two specified sizes is calculated, assuming that the component is made from the same 2024 T3 aluminum alloy.
2
The Walker approach
Dealing with the mean stress effects in SN curves, Walker [7] introduced the concept of an equivalent zero of the 1–γ maximum effective stress, Seq=Smax ΔS γ, where γ was assumed to be a material constant. The same idea can be extended to the SIF range, ΔKeq, which would generate the same FCG rate da/dN as the actual (Kmax, R) combination for R > 0. ΔKeq ¼ Kmax ð1 – RÞγ :
(2)
The material parameter γ varies between 0 and 1.0 (maximum and minimum sensitivity to load ratio, respectively). For example, for γ = 1, ΔKeq=ΔK, which implies that the load ratio R has no effect. Thus, γ can be considered to be an inverse measure of the sensitivity of FCG to the mean load or load ratio. One way, but not the only way, to find γ will be described later. Because ΔK = Kmax(1 – R), a relation between the equivalent and applied SIF range is ΔKeq ¼ ΔKð1 – RÞγ – 1 :
(3)
Let one of the Paris’ parameters for the particular case of R = 0 be denoted as C0, and assume that propagation occurs in a stable region of the FCG curve. Then we get da ðR ¼ 0Þ ¼ C0 ΔK mw : dN
(4)
where “m” is a material parameter and the letter “w” means Walker. According to Eq. (3), for R = 0, ΔKeq=ΔK, and the following relation holds true: da ¼ C0 ½ΔKð1 – RÞγ – 1 mw : dN
(5)
Equation (5) represents the dependence for all load ratios. The surface that best fits the (da/dN, ΔK, R) experimental data can be obtained through a regression analysis that returns (C0, γ, mw). The number of variables that describe FCG is reduced to two, either (da/dN, ΔK) or
Jorge Alberto Rodriguez DURAN et al. Engineering application of the FCG approach under nonzero mean loads
(da/dN, ΔKeq), depending on which of the following forms of Eq. (6) is used: da ¼ CΔK mw , dN da mw ) : ¼ C0 ΔKeq dN
C ¼ C0 ð1 – RÞmw ðγ – 1Þ ) ΔKeq ¼ ΔKð1 – RÞ
γ–1
(6)
The integration in Eq. (1) can now proceed for any load ratio, provided that the material constants are available. If the function ΔK(a) or ΔKeq(a) is not trivial, then numerical integration will be needed. As mentioned above, the Walker relationship is valid for R≥0. Dinda and Kujawski [10] proposed a similar relation for positive and negative load ratios. However, their proposal used the ΔK+ parameter that represents the positive part of the applied SIF, and assumed that the negative part of ΔK does not contribute to FCG.
3
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the load level, geometry, and crack size, thereby preserving the independence of the measured da/dN values of these parameters [12]. In terms of life prediction, the constant da/dN curves according to the UA can be described by a power law, of the form: da * Þq : ¼ AðΔK – ΔKth* Þp ðKmax – Kmax dN
(7)
As in the Walker approach, the constants in Eq. (7) can be determined by a multiple linear regression (MLR) of the experimental data for various load ratios.
The unified approach
To account for discrepancies in relation to plasticity induced crack closure (see above), Sadananda and Vasudevan [8,9] proposed a new concept requiring two driving forces for an advancing fatigue crack: ΔK and Kmax. Two physical reasons motivated their choice: 1) The total damage ahead of a crack tip is the sum of monotonic and cyclic damages due to Kmax and ΔK, respectively; and 2) FCG can only occur if tensile stresses exist in the monotonic plastic zone (i.e., for Kmax > 0). Sadananda and Vasudevan [8,9] collected extensive raw (without crack closure correction) da/dN vs. ΔK data for a wide range of load ratios, and plotted them in the ΔK vs. Kmax coordinates. Regardless of the test method, all of the data for a given FCG rate fell on a single curve, demonstrating that the behavior is unique for a particular material. The authors determined that an intrinsic threshold, corresponding to a given da/dN, exists for each * variable and called them ΔKth* and Kmax,th , respectively. These thresholds increase with increasing FCG rate. For an FCG test at a constant load ratio, the intersection of both thresholds forms the so-called FCG trajectory, which corresponds to different crack growth mechanisms. For a specific material and environment, the FCG mechanism is controlled by Kmax at low load ratios and by ΔK at high FCG rates [11]. According to the unified approach (UA), the load ratio R is not a driving force and does not have a threshold. Consequently, it cannot be used as a second parameter for the FCG. Figure 1 shows a schematic representation of the constant da/dN curves in the ΔK-Kmax, th coordinates and their respective intrinsic thresholds. In this approach, there is no need for an extrinsic factor (e.g., crack closure). The constant da/dN curves fully describe the material behavior, and laboratory data can be directly used in design. Both ΔK and Kmax are LEFM parameters that, by definition, include
Fig. 1 Schematic representation of the constant FCG rates plotted in ΔK-Kmax coordinates, and the intrinsic thresholds according to the UA
4 Numerical fitting of experimental data to the Walker and unified approaches Linear regression is a powerful tool that is widely used by engineers to find the curve (or surface) that best fits experimental data. The process assumes that a linear relation exists between the variables. Linear forms of Eqs. (5) and (7) can be obtained by taking the logarithms of both sides. For the UA, the result is log
da ¼ log A þ plogðΔK – ΔKth* Þ þ ::: dN * Þ, þ qlogðKmax – Kmax
(8)
which represents the equation of a plane in a threedimensional coordinate system, where: y ¼ a0 þ a1 x þ a2 z, da , dN x ¼ logðΔK – ΔKth* Þ,
y ¼ log
* Þ, z ¼ logðKmax – Kmax
a0 ¼ logA,
a1 ¼ p, a2 ¼ q :
(9)
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The same procedure allows Walker’s parameters to be determined, as follows: y ¼ a0 þ a1 x þ a2 z, da , x ¼ logð1 – RÞ, z ¼ logΔK, dN a0 ¼ logC0 , a1 ¼ mw ðγ – 1Þ, a2 ¼ mw :
y ¼ log
(10)
Although intuitive, the idea of including the mean stress corrections into the FCG equations and performing a single fitting procedure was only proposed recently [13]. For the determination of the coefficients in Eqs. (9) and (10) (consequently, the parameters for each approach), a complete set of experimental data (da/dN vs. ΔK) at different load ratios is needed. Over 300 experimental data points (ΔK, da/dN) for the 2024 T3 aluminum alloy at
Fig. 2 alloy
Experimental FCG curves for the 2024 T3 aluminum
positive load ratios ranging from R = 0.1 to R = 0.8 were obtained from tables in Ref. [14]. All of them are plotted here in Fig. 2. Both of the approaches treated in this paper (Walker and UA) require MLR analyses. MLR is an easy and wellestablished process, the mathematical details of which can be found elsewhere [15]. To accomplish this task, a special purpose code is written in the MapleTM software environment. The vectors x, y, and z are initially obtained from experimental data, according to Eqs. (9) and (10). For the * UA, the thresholds are ΔKth* = 1.0 MPa$m1/2 and Kmax =3 1/2 MPa$m , corresponding to the minimum values of the experimental data. These thresholds are in the expected range for aluminum alloys [16]. Figure 3 shows, for both approaches, the plane that best-fitted the set of points (xi, yi, zi), where i is between 1 and the length of the vectors x, y, and z, together with these points. The visual correlation is quite good, and the mathematical coefficient exceeds 99.99% in both cases. Coefficients of these planes allows the parameters for each approach to be determined, in accordance with Eqs. (9) and (10). These results are summarized in Table 1. Through the fitting parameter γ, Walker approach reduces the number of variables that describe the FCG. The graphical success of the correlation in UA can also be visualized in Kmax vs. ΔK coordinates, by plotting the raw experimental FCG data, along with some constant FCG rate curves (Eq. (7)) with parameters from Table 1 (see results in Fig. 4). As expected from the analysis of Fig. 3, good agreement can be found between the experimental data and the fitted curve. Once the parameters of both approaches have been defined, and considering that these parameters already include the load-ratio effects, the designer can calculate the residual life of any component, given a specific material
Fig. 3 Three-dimensional representation of elements of vectors x, y, and z (see Eqs. (9) and (10)), and the plane that best fits them using the (a) Walker and (b) unified approaches
Jorge Alberto Rodriguez DURAN et al. Engineering application of the FCG approach under nonzero mean loads
Table 1
259
Coefficients and parameters of the Walker and unified approaches (Eqs. (5) and (7)), obtained after an MLR analysis of the experimental
data Coefficients Approach
a0
a1
a2
Parameters
Walker
– 10.97
– 1.24
3.95 = mw
C0= 1.06E – 11 m/cycle/(MPa$m1/2)mw, γ = 0.68
Unified
– 9.47
1.97 = p
0.72 = q
A = 1.06E – 11 m/cycle/(MPa$m1/2)p + q
Fig. 4 Some constant FCG rates (in “m” and using Eq. (11) with parameters from Table 1) and the raw experimental FCG data in UA coordinates
Fig. 5 SIF for a pair of cracks emanating from a circular hole in a wide plate under a nominal stress S, and variation of the Fl(s) parameter
and SIF for the particular geometry and type of loading. This issue is addressed below.
material is assumed to be the same (2024 T3 aluminum alloy) for all methods, the FCG experimental data of which are shown in Fig. 2. Parameters from Table 1 are used in connection with each approach. Figure 6 shows the results in Nif vs. sf coordinates, where sf is the final value of the sparameter for each calculation.
5 Numerical life predictions with the Walker and unified approaches Cracks growing from notches are among the most common situations encountered by designers. Here, the case of a pair of cracks emanating from a circular hole in a wide plate is used as a numerical example. The SIF-solution (Fig. 5) for this particular case can be found in the classical handbook of Tada et al. [17]. The solution includes mode I and mode III SIFs, but only mode I (loading normal to the crack surface) is analyzed in this paper. The Fl(s) parameter has a pronounced variation for 0.2 < s < 0.8. The residual life Nif can be determined with Eq. (1) through numerical integration. As stated before, a computational code is written with this aim. In the numerical example, the circular hole had a radius R0= 10 mm (see Fig. 5), and the initial crack size is 2.5 mm (or s = 0.2). The residual life Nif is calculated in the range 0.2 < s < 0.8 (i.e., final crack size = 40 mm). The SIF range (in MPa$m1/2) varied in the interval 3.4 < ΔK < 6.6, and the load ratio is maintained at R = 0.4. Two methods to handle the R-dependence in FCG are simulated. The
Fig. 6 Calculated residual life versus the final value of the sparameter for a crack growing between 2.5 and 40 mm from the circular hole of Fig. 5
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Another FCG simulation is made in a plate (width b = 40 mm, thickness t) with an edge crack and under a nominal stress S. A final value sf= 0.5 is considered sufficient for illustration purposes. This simulation is made under similar load ratio and material conditions as those used for the crack emanating from a hole, with the main difference being the initial (8 mm) and final (20 mm) crack sizes. The range of SIF varies from 3.6 to 11.8 MPa$m1/2 and the results are showed in Fig. 7. Table 2 summarizes the loading conditions used for the FCG simulations in two geometries: Geometry 1, a pair of cracks emanating from a circular hole in a wide plate (Fig. 5), and Geometry 2, an edge crack in a plate (Fig. 7).
region of low da/dN. In curves like those in Fig. 6, Nif is expected to be independent of the final crack size. However, the situation depicted in this figure is the result of the particular behavior of cracks emanating from notches. For small crack sizes, their SIF-solution approximately follows that of a surface crack in a smooth infinite body, except for the use of the notch stress concentration factor for increasing the nominal gross stress. Once these cracks have grown far from the hole, the SIF-solution follows that of a single long crack in a wide plate, and falls well below that for the smooth member. Only a small part of the total life is spent with the crack under the influence of the notch stress fields. In accordance with Fig. 6, the residual life of these types of cracks does not depend primarily on the initial crack size, a fact that facilitates the use of LEFM approaches for stress-life predictions. The second simulation shows the most common trend (Fig. 7) — namely, the greater the final crack size is, the less influence it has on the residual life calculations. This trend appears because these types of cracks have a geometric factor that grows rapidly with the parameter sf. A more formal analysis can be done to explain the differences between the a vs Nif curves for cracks departing from notches (Fig. 6) or growing from smooth surfaces (Fig. 7). First, a dimensionless SIF expression for both types of cracks can be obtained as follows: rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi R K ¼ S πaFðsÞ ¼ S πa 0 FðsÞ, R0 rffiffiffiffiffiffiffiffi K a pffiffiffiffiffi ¼ π FðsÞ: R0 S R0
Fig. 7 Residual life Nif calculated for a plate Table 2 Loading conditions for the FCG simulations in Geometries 1 and 2, with constant load ratio (R = 0.4) Smax /MPa
Initial SIF /(MPa$m1/2)
Final SIF /(MPa$m1/2)
1
28
3.4
6.6
2
28
3.6
11.8
Geometry
6
Discussion
The two methodologies used for the numerical examples — namely, the Walker and unified approaches — overcome the controversial closure problem and its derivate relations. These methods are two-parameter approaches (ΔK and Kmax, instead of ΔK and R) and use surface fitting to find the adjustable constants. Once these constants are found for a given material, the similitude principle can be used to design any component or structure, given only independent expressions of ΔK and R. History effects are not considered because FCG is assumed to occur under mostly linear elastic conditions. By virtue of the classical form of FCG curves, cracks should spend most of their lives in the near-threshold
(11)
Geometric factors for both geometries can be expressed as functions of the dimensionless parameter X≡a/R0, where Geometry 2 is also dependent on the ratio R0/b: 3 – X =ðX þ 1Þ f1 þ 1:243½1 – X =ðX þ 1Þ3 g, 2 FðX ,R0 =bÞ ¼ 0:265ð1 – XR0 =bÞ4 þ :::
FðX Þ ¼
þ
0:857 þ 0:265XR0 =b ð1 – XR0 =bÞ3=2
: (12)
Equations (11) and (12) are plotted in Fig. 8 for small and large b, including R0/b = 1/4 (as used in this paper). The smooth component has no hole, so the graphs should be interpreted only on a comparative basis. For small R0/b, the crack driving force in smooth components (Ks) asymptotically approximates to the SIF in notched components (Kn) when the crack size is much larger than the hole radius. For large values of a/R0, Ks will exceed Kn, and the crack will grow quicker in the smooth than in the notched component. The same outcome occurs for smaller values of a/R0 (near 1.0) and for R0/b = 1/4, but with a steeper curve.
Jorge Alberto Rodriguez DURAN et al. Engineering application of the FCG approach under nonzero mean loads
261
Nomenclature a
Crack size
A, C, m, γ, p, q Material parameters
Fig. 8 Comparison between normalized SIFs for cracks in notched and smooth components. The (a vs. Nif) curve of Fig. 7 corresponds to a smooth component, with width b that is four times the radius of the hole R0 of the notched component (Fig. 6)
The opposite trends in the slope of each (a vs. Nif) curve (Figs. 6 and 7) can now be easily understood. Due to the stress concentration and the relative proximity of the free surface in the smooth component (high R0/b), Kn is greater than Ks in the short crack regime, but K(a) and, consequently, the FCG rates are always growing in the smooth component. The opposite situation exists in the notched component, where K(a) and da/dN both decrease. As a result, the pair of cracks emanating from the hole grows continuously, but at a decreasing rate in the analyzed s-range (Fig. 6). Cracks growing from the smooth surface spend most of their lives in the near-threshold (low ΔK) region. The relative position of the curves in Fig. 8 is purely geometric, as can be deduced from the generating equations (Eqs. (11) and (12)). In both simulations, higher FCG rates are predicted by the UA. However, the thresholds for this approach are chosen as the minimum values among the available experimental * data (ΔKth* = 1.0 MPa$m1/2; Kmax = 3 MPa$m1/2) and are not measured by specific tests.
7
Conclusions
In this paper, two methods used to reduce the dependence of the FCG problem on pure LEFM parameters were numerically tested in the whole sense, namely, from the process of fitting parameters to the residual life calculations. The two methods provided similar results, with the UA returning more conservative results. Designing against the FCG based only in LEFM parameters has some advantages: No consideration about closure is needed, and the load level, geometry, and crack size are all included, thus facilitating the application of the similitude principle.
da/dN
fatigue crack growth rate
F(s)
Geometric factor as a function of normalized crack size
Nf
Total fatigue life to failure, in number of cycles
Ni
Crack initiation life, in number of cycles
Nif
Number of fatigue cycles spent in the propagation stage
P
Load
R
Ratio of the minimum to maximum load
R0
Radius of the hole
s
Normalized crack size, a/b or a/(a + R0)
Ks or Kn
Stress intensity factors, SIFs, for cracks in smooth and notched components, respectively
ΔK
SIF range
S
Nominal stress
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