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International Journalof Fatigue

International Journal of Fatigue 30 (2008) 1967–1977

www.elsevier.com/locate/ijfatigue

New method for evaluation of the Manson–Coﬃn–Basquin and Ramberg–Osgood equations with respect to compatibility Adam Niesłony a,*, Chalid el Dsoki b, Heinz Kaufmann c, Peter Krug d b

a Department of Mechanics and Machine Design, Opole University of Technology, ul. Mikolajczyka 5, 45-271 Opole, Poland Chair of System Reliability and Machine Acoustics (SzM), TU-Darmstadt, Magdalenenstrasse 4, 64289 Darmstadt, Germany c Fraunhofer-Institute for Structural Durability and System Reliability, LBF, Bartningstrasse 47, 64289 Darmstadt, Germany d PEAK Werkstoﬀ GmbH, Siebeneickerstrasse 235, 42553 Velbert, Germany

Received 5 July 2007; received in revised form 6 November 2007; accepted 10 January 2008 Available online 1 February 2008

Abstract In the modern description of the fatigue behaviour of materials the stress–strain curve, described with Ramberg–Osgood equation, and the strain–life curve, described with Manson–Coﬃn–Basquin equation, are typically used. It is known that the assumption of equality of the plastic and elastic components in both equations leads to the so-called compatibility condition and connect the equations theoretically. The conventional method for evaluation of the fatigue parameters use one set of experimental data from strain-controlled uni-axial fatigue tests but they not ensure the compatibility conditions. The presented new method for determining the stress–strain and strain–life curves retains the mathematical and physical relationships between the considered curves. The method involves ﬁtting the curve to experimental data points in a three-dimensional strain–stress–life space. With the plastic part of strain, stress and fatigue life as coordinates, a straight line is used for ﬁtting the experimental data points. The material parameters are calculated directly from projections of the three-dimensional straight line on suitable planes. The results obtained from this new method using high-strength aluminium alloys subjected to diﬀerent manufacturing conditions and diﬀerent test temperatures are presented. These results are then compared to results obtained with a conventional method for determining the fatigue parameters. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Manson–Coﬃn–Basquin curve; Ramberg–Osgood curve; Cyclic fatigue properties; Stress; Strain

1. Introduction Manson and Coﬃn [1–3] were the ﬁrst researchers which related the number of cycles to crack initiation to the amplitude of plastic strain. Together with the modiﬁed form of the Basquin equation [1,4], this work led to the known equation of the strain–life curve, Fig. 1, where the total strain amplitude ea,t is divided into elastic ea,e, and plastic ea,p components as such: r0 b c ea;t ¼ ea;e þ ea;p ¼ f ð2N f Þ þ e0f ð2N f Þ : ð1Þ E *

Corresponding author. Tel.: +48 (0) 77 40 06 379; fax: +48 (0) 77 40 06 343. E-mail address: [email protected] (A. Niesłony). 0142-1123/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2008.01.012

The Young’s modulus E is obtained either from a static tensile material test or from the ﬁrst hysteresis loop recorded during the fatigue test. To determine the remaining four material constants, it is necessary to perform several fatigue tests under cyclic loading with constant amplitude and a load ratio of Re = 1 on unnotched specimens. Due to hardening or softening eﬀects, materials subjected to cyclic loading are often unstable, producing stress amplitude changes during the tests [5,6]. Thus, the stress amplitude corresponding to the stabilized state must be used, which occurs at the half of the number of cycles to crack initiation, which was determined at 10% stiﬀness loss of the stiﬀness over cycles signal. With this stress amplitude, the Young’s modulus and the total strain amplitude the plastic part of the strain amplitude can be calculated

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Nomenclature ea,t, ea,e, ea,p total, elastic and plastic strain amplitude, respectively ra stress amplitude e0 plastic strain limit value e0f fatigue ductility coeﬃcient r0f fatigue strength coeﬃcient b fatigue strength exponent c fatigue ductility exponent E Young’s modulus (modulus of elasticity) K0 cyclic strength coeﬃcient n0 cyclic strain hardening exponent 2Nf number of cycles to fatigue failure (crack initiation) R directional vector

l, m, n Re R0p0:2 V K 0 ; V n0

direction cosines of the vector R strain ratio yield limit for 0.2% residual elongation compatibility ratio for K0 and n0 , respectively

Subscripts comp. calculated from compatibility conditions or compatibility is given conv. conventional method m measured Superscripts MCB, RO Manson–Coﬃn–Basquin or Ramberg–Osgood, respectively

Fig. 1. Example of the Manson–Coﬃn–Basquin curve according to Eq. (1).

ea;p ¼ ea;t ea;e ¼ ea;t

ra : E

ð2Þ

Another possibility for determining the plastic part of the strain amplitude is to use the measured value of the thickness of the recorded stable hysteresis loop [5,7], Fig. 2a. However, for materials in which the linear part of the decreasing hysteresis curve becomes non-linear before reaching the zero-stress level, Fig. 2b, the value of measured plastic strain ea,p,m and computed ea,p by Eq. (2) are unequal, particularly when the specimen is subjected to higher strain amplitudes. To guarantee the comparability of the cyclic parameters found in literature [9], this study uses Eq. (2) for evaluating the plastic strain [1,5,8]. To compute the material constant in Eq. (1) from experimental data the least squares method for ﬁtting the component curves can be applied [10,11]. Linearization of the plastic part of Eq. (1) leads to Y ¼ logðe0f Þ þ cX ;

ð3Þ

where Y = log (ea,p) and X = log (2Nf). While ﬁtting the line, Eq. (3), only those points should be take into account where the value of the plastic strain is higher than the established limit e0. This limitation results from the measuring accuracy of the total strain and from the operations of subtraction in Eq. (2). For steels, the value e0 = 0.01% is recommended [12]. Many problems related to the strain limit e0 have been discussed in [10]. Linearization of the elastic part of Eq. (1) can be done in the same manner 0 r Y ¼ log f þ bX ; ð4Þ E where Y = log (ea,e), and X = log (2Nf). Ramberg and Osgood [1,13] proposed a method of describing stresses and strains in materials subjected to constant amplitude cyclic loading. This method is widely applied in fatigue analyzes as an important source of information about the material behaviour [5,7]. Like in the

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

a

b

σ

1969

σ

εa,p,m = εa,t – εa,e

εa,p,m = εa,t – εa,e

ε

εa,p,m

ε

εa,e

εa,p,m

εa,e

Fig. 2. Calculated (a) and measured (b) plastic parts of the strain amplitude.

Manson–Coﬃn–Basquin equation, the total strain amplitude is divided into elastic and plastic components, but described as a function of stress amplitude ra and not fatigue life Nf ra ra n10 ea;t ¼ ea;e þ ea;p ¼ þ : ð5Þ E K0 The ﬁrst term of the equation is linear (the elastic component), and the second term is logarithmic (the plastic component). The stress–strain curve is usually presented in a linear coordinate system, Fig. 3a, but in the double-logarithmic coordinate system, the terms of the equations are expressed by straight lines, Fig. 3b. Thus, in order to determine the material parameters K0 and n0 , the plastic part of the Eq. (5) must be linearized. Linearization is performed by ﬁnding the logarithms of the equation 0

ra ¼ K 0 ena;p ;

ð6Þ

and thus obtaining

Y ¼ logðK 0 Þ þ n0 X ;

ð7Þ

where Y = log (ra), and X = log (ea,p). The experimental points are obtained from the constant amplitude fatigue tests and ﬁtted using the least-squares method. This approximation only takes those points into account where the plastic strain is greater than the assumed limit value e0. Here, the test results used for determining the Manson–Coﬃn–Basquin curve are usually applied [9]. Since many materials stabilize after a relatively short time, it is possible to obtain the experimental points using a more eﬃcient method that uses only one specimen. These methods are: incremental step test, multiple steps with increasing strain, multiple steps with decreasing strain and a method using random cyclic strain amplitude tests [14,15]. The experimental results (pairs of amplitudes of total stress and strain) are considered in the same way as those obtained from constant amplitude tests for stabilised hysteresis, and are ﬁtted as described above. However, the results obtained

Fig. 3. Example of the Ramberg–Osgood curve according to Eq. (5): (a) linear scale; (b) logarithmic scale.

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for materials with unstable material behaviour can diﬀer extensively depending on the used method [9,14]. 2. The compatibility terms Eqs. (1) and (5) have similar structures in that both total strain amplitudes consist of the sum of plastic and elastic parts. By comparing the elastic components, the following equations can be derived RO eMC a;e ¼ ea;e ; r0f

ra ; E E logðr0f Þ þ logð2N f Þb ¼ logðra Þ b

ð2N f Þ ¼

ð8Þ

and for the plastic parts of Eqs. (1) and (5) RO eMC a;p ¼ ea;p ; r 10 a n ¼ e0f ð2N f Þc ; K0

ð9Þ

1 ¼ logðe0f Þ þ logð2N f Þc: n0 Introducing the log (ra) part from Eq. (8) into (9) leads to the following expression 1 logðr0f Þ þ logð2N f Þb logðK 0 Þ 0 ¼ logðe0f Þ þ logð2N f Þc n ð10aÞ 1 1 1 logðr0f Þ 0 þ logð2N f Þ b 0 c logðK 0 Þ 0 ¼ logðe0f Þ ð10bÞ n n n ½logðra Þ logðK 0 Þ

Material parameters as well as number of cycles appear in Eq. (10b). This equation shows three terms containing only material parameters which do not depend on the number of cycles. However, this equation should be valid for the entire range of cycles, which then leads to 1 1 logð2N f Þ b 0 c ¼ 0 if; and only if; b 0 c ¼ 0; ð11Þ |ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ} n n

assumed that the material is stable during fatigue tests and does not exhibit signiﬁcant softening or hardening eﬀects. Therefore, the constants E, K0 , n0 , e0f , r0f , b and c do not depend on the number of cycles Nf or on the values of strain ea or stress ra. This assumption may be invalid for some materials. 3. Problems when compatibility is not ensured The known methods for determining material constants do not ensure compatibility. The material constants are usually determined separately, and the sets of data points are treated as separate experimental results. If Eqs. (12) and (14) are not adhered to, a loss of equality between the plastic and elastic parts occurs in both formulations. To make the problem clear the ratios of the elastic, respectively of the plastic strain over the total stress amplitude were computed according to the ﬂow chart showed in Fig. 4 and presented in Fig. 5. When compatibility is given, the ratio of the elastic part of the strain and the ratio of the plastic part of the strain of the Ramberg–Osgood and the Manson–Coﬃn–Basquin equations is always one for all total strain amplitudes, and therefore for whole range of cycles. Lack of compatibility causes the ratios to deviate from one and yields to non-linear curves. This is incongruent because each tested specimen possesses elastic and plastic strain parts independent of whether it is used for the analysis of the strain–life curve or the stress–strain curve [1,5]. The component terms should be equal in both formulations. When compatibility is not given the line of the ratio of the elastic part deviates from one to higher amplitudes, and the line of the ratio of the plastic part shows large deviations to smaller amplitudes.

6¼0

b n0 ¼ ¼ n0comp: c Substitution of Eq. (12) into (9) leads to 1 1 logðr0f Þ 0 logðK 0 Þ 0 ¼ logðe0f Þ n n r0f 0n0 ¼ ef K0 K0 ¼

ð12Þ

ð13Þ

r0f

0 ð14Þ b ¼ K comp: ðe0f Þc Eqs. (12) and (14) are the so-called compatibility equations and they couple the coeﬃcients and exponents which appear in the Ramberg–Osgood and Manson–Coﬃn–Basquin formulations. This yields the possibility of determining the coeﬃcient and exponent in the stress–strain curve (5) directly from the strain–life curve (1). In Eq. (10) it is assumed, that the number of cycles to failure does not inﬂuence the elastic–plastic properties of the material. Thus, it is

Fig. 4. The ﬂow chart for computing the numerical quantities used in Figs. 5 and 6.

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

1971

Fig. 5. Computed ratios for elastic and plastic strains by neglecting the compatibility.

Thus, any of the following applications connecting the parts of Eqs. (1) and (5) in further calculations can lead to large deviations in the fatigue life of a component. This can be shown if the number of cycles to the fatigue crack occurrence is determined, either by using Eq. (1), or by using the stress amplitude and the following equation: 1 1 ra b Nf ¼ : ð15Þ 2 r0f

If compatibility is satisﬁed Eqs. (15) and (1) leads to the same number of cycles. The consecutive computational steps are shown in Fig. 4 and the resulted values on Fig. 6. The numbers of cycles Nf and N MCB obtained by f way of the two calculations methods are diﬀerent if the compatibility equations are not satisﬁed. That means that the equation leads to two diﬀerent stress amplitudes for one number of cycles or it leads to two diﬀerent number of cycles for one stress amplitude, see Fig. 6. Moreover,

Fig. 6. Calculated number of cycles by neglecting the compatibility.

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A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

the numerically determined stress–life characteristic MCB ðrRO Þ does not produce a straight line in the doua Nf ble-logarithmic coordinate system, thus disagreeing with Eq. (15) used for its description. This can cause signiﬁcant deviation in the determination of the cycle numbers, when the ﬁnal formulae for determining fatigue life are derived on the assumption that the elastic and plastic parts in Eqs. (1) and (5) are equal.

z

z1 P1 ( x1 , y1 , z1 )

R (l , m , n )

4. The proposed method From Eqs. (1) and (5) it appears, that one value of the total strain amplitude, ea,t, corresponds to only one stress amplitude, ra, and to a single number of cycles, Nf. In such a case, it is possible to formulate the total characteristics expressed by the curve (ea,t–ra–Nf) in a three-dimensional space. Fig. 7a shows such characteristics also depicting curves for the parts of elastic strain (ea,e–ra–Nf) and plastic strain (ea,p–ra–Nf). By a projection of the three-dimensional characteristics, the following curves can be obtained: projection to the plane (ea,t–ra) gives the trajectory of the cyclic stress–strain curve (Ramberg–Osgood equation), projection to the plane (ea,t–Nf) gives the strain–life curve (Manson–Coﬃn–Basquin equation), and projection to the plane (ra–Nf) gives the stress–life curve, expressed with Eq. (15). Particular projections for 3-D characteristics are shown in Fig. 7b–d.

y1

y

x1 x Fig. 8. Points in the three-dimensional space with the regression line determined by point P1 and directional vector R.

In this new method, all six cyclic characteristics are determined through the approximation of the regression line in the 3-D space, and there is no dependence on the regression direction. The proposed method retains compatibility. Location of the straight line (ea,p–ra–Nf) in 3-D space can be deﬁned by coordinates of the point P1(x1, y1, z1) located on this straight line and the directional vector R(l, m, n) to which the straight line is parallel [15], Fig. 8.

Fig. 7. (a) Three-dimensional space with the strain–stress–life curve (ea–ra–Nf) and its projections on (b) strain–stress (ea–ra) plane, (c) stress–life (ra–Nf) plane, and (d) strain–life (ea–Nf) plane.

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

It is proposed to approximate the points with the least-square method, minimizing the sum of the squares of the distances of the points from the straight line. There are many eﬃcient methods for such regressions in the 3-D space [16,17]. The location of the straight line in 3-D space also determines the positions of its projections on the particular planes. Thus, the material parameters of interest can be determined directly from the following equations: m ; l 0 K 0 ¼ 10ðy 1 x1 n Þ ;

n0 ¼

l m c¼ ; b¼ ; n n e0f ¼ 10ðx1 z1 cÞ ; r0f ¼ 10ðy 1 z1 bÞ :

ð16Þ

These equations have been derived according to the following assumptions: x ¼ logðea;p Þ;

y ¼ logðra Þ;

z ¼ logðN f Þ:

ð17Þ

As in the conventional method, the established limit e0 is considered in the 3-D method. While ﬁtting the line, only those points were considered where the value of plastic strain is higher than e0. However, it is not recommended to do this directly since omission of such points can greatly inﬂuence the stress–life curve (ra–Nf), where the elastic part of the strain plays a signiﬁcant role. Therefore, it is proposed to recalculate the plastic strain parts with e0 < 0.01% so that the new values do not change the strain–life curve (ea–Nf). These new values then lie directly on the line of the plastic part of the Ramberg–Osgood equation, which was determined without considering these points. However, the stress values which form the third plane of these tests are not corrected, and are subject to further scatter. Subsequently, a second regression is carried

1973

out in order to recalculate the location of the (ea,p–ra–Nf) line. The algorithm of the computations is shown in Fig. 9. 5. Practical implementation The proposed method was used for calculating the fatigue parameters from Ramberg–Osgood and Manson–Cofﬁn–Basquin equations for spray-compacted aluminium alloys (DISPAL = DISPersion hardened ALuminium) made by PEAK Werkstoﬀ GmbH. These materials are characterized by their high Young’s modulus, good wear resistance and a low coeﬃcient of thermal expansion. In the framework of a bilateral project, several tests on diﬀerent materials were carried out by varying the test conditions. A detailed description of the materials and the static and fatigue tests can be found in previous studies [18,19]. For each material, the constant strain amplitude tests were performed under alternating load and the numbers of cycles to crack initiation were recorded. The stress amplitude was obtained from the recorded stable hysteresis loop at half of the number of cycles to crack initiation. For comparison purposes, the conventional method of determining the fatigue parameters [18,19] was also calculated according to the algorithm shown in Fig. 7. The obtained results can be found in Tables 1 and 2. Since the conventional method does not satisfy the compatibility relations, two simple parameters, V K 0 and V n0 , were deﬁned V K0 ¼

K0 K 0comp:

;

V n0 ¼

n0 n0comp:

:

ð18Þ

These parameters show the diﬀerences between material constants K0 and n0 obtained from Eqs. (12),(14), and

Fig. 9. Algorithms for calculation of the material constants K0 , n0 ; e0f , c, r0f and b.

1974

Number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 21 22 23 24

Material and heat treatment

Pressing

S225 F S250 F S220 F S260 F S250 F S250 F S250 F S226 T6 S250 F S250 F S250 F S230B T6 S232 T6 S232 T6 S691 T6 S693 T6 S691 T6 S693 T6

19 16 13 7A 15 29 34 5B 34 28 30 6A 2 2 7 40 7 40A

Raw part diameter (mm)

Position in the raw part

Temperature (°C)

Young’s modulus E (GPa)

Cyclic values according to Manson–Coﬃn–Basquin curve Fatigue strength coeﬃcient (MPa) r0f Conv. 3-D

Fatigue strength exponent b Conv. 3-D

Fatigue ductility coeﬃcient (m/m) e0f Conv. 3-D

Fatigue ductility exponent c Conv. 3-D

Fitting quality (%) R2 Conv. 3-D

30 30 30 30 30 60 100 30 100 60 60 30 30 30 30 30 30 30

l l l l l l l l c l l l l l l l l l

20 20 20 20 200 20 20 20 20 20 20 20 20 200 20 20 200 200

100.0 100.0 92.5 90.0 90.0 100.0 100.0 92.5 100.0 97.5 97.5 92.5 92.5 86.0 75.0 73.0 73.0 71.0

241.4 394.0 228.1 420.6 284.0 455.0 330.3 499.5 369.2 407.9 370.5 411.4 654.9 450.0 1030.5 866.6 499.7 464.6

0.063 0.099 0.082 0.061 0.073 0.110 0.075 0.063 0.082 0.100 0.088 0.063 0.066 0.086 0.115 0.098 0.082 0.077

0.046 0.035 0.121 0.100 0.038 0.048 0.038 0.019 0.029 0.049 0.035 0.224 0.069 1.219 10.054 15.781 1.358 0.573

0.431 0.359 0.500 0.620 0.324 0.374 0.356 0.484 0.333 0.369 0.330 0.668 0.739 0.847 1.237 1.445 0.851 0.765

97.6 93.8 96.6 98.4 98.3 98.6 94.6 97.1 99.2 98.9 97.7 86.7 97.3 96.8 99.3 97.8 97.1 98.6

241.6 396.9 228.7 420.7 284.1 455.5 331.6 499.9 369.5 408.3 371.3 413.4 655.1 449.8 1030.6 867.4 499.8 463.9

0.063 0.099 0.082 0.061 0.073 0.110 0.075 0.063 0.082 0.100 0.088 0.063 0.066 0.086 0.115 0.099 0.082 0.076

0.046 0.036 0.124 0.107 0.038 0.048 0.039 0.019 0.029 0.049 0.035 0.244 0.072 1.482 10.945 36.385 1.504 0.605

0.432 0.361 0.502 0.628 0.324 0.374 0.358 0.487 0.333 0.369 0.331 0.678 0.744 0.868 1.247 1.557 0.862 0.772

97.5 93.8 96.4 98.3 98.4 98.6 94.8 97.0 99.2 98.9 97.7 85.4 97.3 96.2 99.2 97.5 96.9 98.6

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

Table 1 Cyclic values according to the Manson–Coﬃn–Basquin curve

Number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 21 22 23 24

Material and heat treatment

S225 F S250 F S220 F S260 F S250 F S250 F S250 F S226 T6 S250 F S250 F S250 F S230B T6 S232 T6 S232 T6 S691 T6 S693 T6 S691 T6 S693 T6

Pressing

19 16 13 7A 15 29 34 5B 34 28 30 6A 2 2 7 40 7 40A

Raw part diameter (mm)

30 30 30 30 30 60 100 30 100 60 60 30 30 30 30 30 30 30

Position in the raw part

l l l l l l l l c l l l l l l l l l

Temperature (°C)

20 20 20 20 200 20 20 20 20 20 20 20 20 200 20 20 200 200

Young’s modulus E (GPa)

100.0 100.0 92.5 90.0 90.0 100.0 100.0 92.5 100.0 97.5 97.5 92.5 92.5 86.0 75.0 73.0 73.0 71.0

Cyclic values according to Ramberg–Osgood curve Cyclic hardening coeﬃcient (MPa) K0 Conv. 3-D K 0Comp: 393.7 1009.3 380.2 465.3 590.4 1099.0 649.1 730.4 877.0 923.7 912.2 464.1 956.3 248.6 675.3 572.5 273.3 333.7

379.4 991.4 322.2 523.4 593.8 1108.5 654.0 835.9 879.9 924.9 902.3 471.6 828.1 432.5 826.3 691.0 480.9 487.6

380.0 988.5 322.7 528.4 594.1 1109.8 654.5 839.5 880.1 925.0 901.0 473.6 832.0 441.0 831.2 718.1 485.2 491.3

V K0

Cyclic hardening exponent n0 Conv. 3-D n0Comp:

V n0

Fitting quality (%) R2 Conv. 3-D Comp.

1.036 1.021 1.178 0.881 0.994 0.990 0.992 0.870 0.997 0.999 1.012 0.980 1.149 0.564 0.812 0.797 0.563 0.679

0.152 0.277 0.188 0.082 0.225 0.292 0.208 0.112 0.245 0.271 0.267 0.091 0.105 0.025 0.066 0.041 0.016 0.047

1.036 1.011 1.144 0.832 0.996 0.995 0.995 0.857 0.998 0.999 1.007 0.966 1.176 0.248 0.709 0.597 0.170 0.474

95.5 99.2 99.3 99.6 99.6 99.7 97.9 99.8 98.7 99.7 99.7 96.9 99.9 98.5 99.9 99.9 99.1 96.0

0.147 0.275 0.164 0.098 0.226 0.293 0.209 0.130 0.246 0.271 0.266 0.093 0.089 0.099 0.092 0.063 0.095 0.099

0.147 0.274 0.164 0.099 0.226 0.293 0.209 0.130 0.246 0.271 0.266 0.094 0.089 0.102 0.093 0.068 0.096 0.100

95.4 99.2 99.0 99.5 99.6 99.7 97.9 99.7 98.7 99.7 99.6 96.9 99.7 97.2 99.7 99.7 97.3 94.9

95.4 99.2 99.0 99.4 99.6 99.7 97.9 99.7 98.7 99.7 99.6 96.9 99.7 97.1 99.7 99.7 97.3 94.9

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

Table 2 Cyclic values of the Ramberg–Osgood curve

1975

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A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

Fig. 10. Stress–strain curves obtained with the conventional and the 3-D method.

Fig. 11. Strain–life curves obtained with the conventional and the 3-D method.

(16). The stress–strain and strain–life curves for three selected materials are shown in Figs. 10 and 11. This new method of evaluation will now also be used in the context of the Collaborative Research Centre CRC 666 project ‘‘Integral Sheet Metal Design with Higher Order Bifurcation” (started 2005). The aim of two subprojects in the CRC 666 at the chair of System Reliability and Machine Acoustic SzM of the Technical University of Darmstadt is to analyze and evaluate material characteristics resulting from the new manufacturing method called

‘‘linear ﬂow splitting” [20,21]. With this new processing method, integral design proﬁles are produced by utilising very high degrees of deformation and strong surface hardening eﬀects. These proﬁles are aﬄicted with gradients (residual stress, hardness and microstructure) that require exact knowledge of the local conditions. Here, it was important to closely analyze the compatibility conditions and the methods for evaluating the strain–life curve and the stress–strain curve such that diﬀerences in the stress– strain and the strain–life curve are coming from diﬀerent

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

existing local condition and not from the evaluation method. 6. Conclusion The proposed 3-D method uses a spatial approximation of the test results in order to determine the cyclic parameters. The method yields complete compatibility. The straight line is determined by the least-squares analysis in the 3-D space and the accuracy of the regression is indicated. The conventional methods depended on the regression line (3), (4), or (7), and on the direction (x-axis or y-axis) in which the regression was carried out. This dependence had a large inﬂuence on the cyclic coeﬃcients and exponents, and compatibility was not always given because the assumptions did not allow such dependences. In contrast, the new 3-D analysis method guarantees compatibility. However, for certain materials, it cannot describe the stress–strain curve adequately. For various materials, the determined stress–strain curve derived from the compatibility condition does not agree with the test results, and therefore the materials do not possess compatibility behaviour. Describing this material behaviour is one object of further investigation in the framework of the CRC 666 and a method will be presented soon. Acknowledgements The introduced results were developed in the framework of the German research foundation in the subproject C2 of the CRC 666, entitled ‘‘Developing Methods for the Estimation of the structural durability and System Reliability of Integrated Construction of Metal Sheets”. Furthermore, the authors thank the PEAK Material GmbH in Velbert, Germany, for the provision of material data through which the applicability of this analysis method could be shown. Dr. Adam Nieslony is pleasured to acknowledge the Alexander von Humboldt Foundation for sponsoring his stay in Germany. References [1] Mitchell MR. Fundamentals of modern fatigue analysis for design. In: Lampman Steven R, editor. ASM handbook. Materials Park: ASM International; 1996. p. 229–49.

1977

[2] Manson SS. Fatigue: a complex subject – some simple approximation. Exp Mech 1965;5:193–226. [3] Coﬃn LF. A study of the eﬀect of cyclic thermal stresses on a ductile metal. Trans ASME 1954;76:931–50. [4] Basquin OH. The exponential law of endurance tests. Am Soc Test Mater Proc 1910;10:625–30. [5] Rice RC, Leis BN, Berns HD, Nelson DV, Lingenﬂeser D, Mitchell MR. Fatigue design handbook, Warrendale: SAE; 1988. p. 369. [6] Plumtree A, Abdel-Raouf HA. Cyclic stress–strain response and substructure. Int J Fatigue 2001;23:799–805. [7] Skelton RP, Maier HJ, Christ H-J. The Bauschinger eﬀect, Masing model and the Ramberg–Osgood relation for cyclic deformation in metals. Mater Sci Eng A 1997;238:377–90. [8] ASTM Standard E606-92: Standard practice for strain-controlled fatigue testing. In: Annual book of ASTM standards, vol. 03.01. ASTM; 1997, p. 523–37. [9] Ba¨umel A, Seeger T. Material data for cyclic loading. Mater Sci Monogr 1990;61(Suppl. 1). Amsterdam: Elsevier Science Publishers. [10] Williams CR, Lee Y-L, Rilly JT. A practical method for statistical analysis of strain–life fatigue data. Int J Fatigue 2003;25: 427–436. [11] Williams CR, Lee Y-L, Rilly JT. Characterizing statistical strain–life fatigue parameters. Int J Mater Prod Technol 2004;20(Part 1/3): 220–30. [12] Testing and Documentation Guideline for the Experimental Determination of Mechanical Properties of Steel Sheets for CAE-Calculation, STAHL-EISEN-Pru¨fbla¨tter (SEP) des Stahlinstituts VDEh, SEP 1240, 1st ed. Verein Deutscher Eisenhu¨ttenleute VDEh, Du¨sseldorf, 2006; p. 64 [in German]. [13] Ramberg W, Osgood WR. Description of stress–strain curves by three parameters, Technical Note No. 902, National Advisory Committee for Aeronautics, Washington DC, 1943. [14] Jones A, Hudd RC. Cyclic stress–strain curves generated from random cyclic strain amplitude tests. Int J Fatigue 1999;21: 521–30. [15] Landgraf RW, Morrow J, Endo T. Determination of the cyclic stress– strain curve. J Mater JMLSA 1969;4(1):176–88. [16] Bronstein IN, Semendjajew KA. Handbook of mathematics. Berlin: Springer; 2004. p. 1157. [17] Matlab user’s guide, MathWorks Inc., Version 7, 2006, p. 311. [18] El Dsoki C, Krug P, Kaufmann H. Spray compacted high performance aluminium alloys, Konstruktion, vol. 10. VDI-Verlag; 2006. p. 13–14 [in German]. [19] El Dsoki C, Landersheim V, Kaufmann H, Bruder T, Krug P, Bohn T, et al. The fatigue behaviour of spray compacted aluminium materials. Material properties for component design data: from specimen to component. Materialwiss. Werkstoﬀtech. 2006;37(12): 1006–17. [20] Groche P, Vucic D. Multi-chambered proﬁles made from highstrength sheets. In: Production engineering, vol. XIII/1. Annals of the WGP, Hannover; 2006. p. 67–70. [21] Groche P, Vucic D, Jo¨ckel M. Basics of linear ﬂow splitting. J Mater Process Technol 2007;183(2–3):249–55.

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International Journalof Fatigue

International Journal of Fatigue 30 (2008) 1967–1977

www.elsevier.com/locate/ijfatigue

New method for evaluation of the Manson–Coﬃn–Basquin and Ramberg–Osgood equations with respect to compatibility Adam Niesłony a,*, Chalid el Dsoki b, Heinz Kaufmann c, Peter Krug d b

a Department of Mechanics and Machine Design, Opole University of Technology, ul. Mikolajczyka 5, 45-271 Opole, Poland Chair of System Reliability and Machine Acoustics (SzM), TU-Darmstadt, Magdalenenstrasse 4, 64289 Darmstadt, Germany c Fraunhofer-Institute for Structural Durability and System Reliability, LBF, Bartningstrasse 47, 64289 Darmstadt, Germany d PEAK Werkstoﬀ GmbH, Siebeneickerstrasse 235, 42553 Velbert, Germany

Received 5 July 2007; received in revised form 6 November 2007; accepted 10 January 2008 Available online 1 February 2008

Abstract In the modern description of the fatigue behaviour of materials the stress–strain curve, described with Ramberg–Osgood equation, and the strain–life curve, described with Manson–Coﬃn–Basquin equation, are typically used. It is known that the assumption of equality of the plastic and elastic components in both equations leads to the so-called compatibility condition and connect the equations theoretically. The conventional method for evaluation of the fatigue parameters use one set of experimental data from strain-controlled uni-axial fatigue tests but they not ensure the compatibility conditions. The presented new method for determining the stress–strain and strain–life curves retains the mathematical and physical relationships between the considered curves. The method involves ﬁtting the curve to experimental data points in a three-dimensional strain–stress–life space. With the plastic part of strain, stress and fatigue life as coordinates, a straight line is used for ﬁtting the experimental data points. The material parameters are calculated directly from projections of the three-dimensional straight line on suitable planes. The results obtained from this new method using high-strength aluminium alloys subjected to diﬀerent manufacturing conditions and diﬀerent test temperatures are presented. These results are then compared to results obtained with a conventional method for determining the fatigue parameters. Ó 2008 Elsevier Ltd. All rights reserved. Keywords: Manson–Coﬃn–Basquin curve; Ramberg–Osgood curve; Cyclic fatigue properties; Stress; Strain

1. Introduction Manson and Coﬃn [1–3] were the ﬁrst researchers which related the number of cycles to crack initiation to the amplitude of plastic strain. Together with the modiﬁed form of the Basquin equation [1,4], this work led to the known equation of the strain–life curve, Fig. 1, where the total strain amplitude ea,t is divided into elastic ea,e, and plastic ea,p components as such: r0 b c ea;t ¼ ea;e þ ea;p ¼ f ð2N f Þ þ e0f ð2N f Þ : ð1Þ E *

Corresponding author. Tel.: +48 (0) 77 40 06 379; fax: +48 (0) 77 40 06 343. E-mail address: [email protected] (A. Niesłony). 0142-1123/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2008.01.012

The Young’s modulus E is obtained either from a static tensile material test or from the ﬁrst hysteresis loop recorded during the fatigue test. To determine the remaining four material constants, it is necessary to perform several fatigue tests under cyclic loading with constant amplitude and a load ratio of Re = 1 on unnotched specimens. Due to hardening or softening eﬀects, materials subjected to cyclic loading are often unstable, producing stress amplitude changes during the tests [5,6]. Thus, the stress amplitude corresponding to the stabilized state must be used, which occurs at the half of the number of cycles to crack initiation, which was determined at 10% stiﬀness loss of the stiﬀness over cycles signal. With this stress amplitude, the Young’s modulus and the total strain amplitude the plastic part of the strain amplitude can be calculated

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Nomenclature ea,t, ea,e, ea,p total, elastic and plastic strain amplitude, respectively ra stress amplitude e0 plastic strain limit value e0f fatigue ductility coeﬃcient r0f fatigue strength coeﬃcient b fatigue strength exponent c fatigue ductility exponent E Young’s modulus (modulus of elasticity) K0 cyclic strength coeﬃcient n0 cyclic strain hardening exponent 2Nf number of cycles to fatigue failure (crack initiation) R directional vector

l, m, n Re R0p0:2 V K 0 ; V n0

direction cosines of the vector R strain ratio yield limit for 0.2% residual elongation compatibility ratio for K0 and n0 , respectively

Subscripts comp. calculated from compatibility conditions or compatibility is given conv. conventional method m measured Superscripts MCB, RO Manson–Coﬃn–Basquin or Ramberg–Osgood, respectively

Fig. 1. Example of the Manson–Coﬃn–Basquin curve according to Eq. (1).

ea;p ¼ ea;t ea;e ¼ ea;t

ra : E

ð2Þ

Another possibility for determining the plastic part of the strain amplitude is to use the measured value of the thickness of the recorded stable hysteresis loop [5,7], Fig. 2a. However, for materials in which the linear part of the decreasing hysteresis curve becomes non-linear before reaching the zero-stress level, Fig. 2b, the value of measured plastic strain ea,p,m and computed ea,p by Eq. (2) are unequal, particularly when the specimen is subjected to higher strain amplitudes. To guarantee the comparability of the cyclic parameters found in literature [9], this study uses Eq. (2) for evaluating the plastic strain [1,5,8]. To compute the material constant in Eq. (1) from experimental data the least squares method for ﬁtting the component curves can be applied [10,11]. Linearization of the plastic part of Eq. (1) leads to Y ¼ logðe0f Þ þ cX ;

ð3Þ

where Y = log (ea,p) and X = log (2Nf). While ﬁtting the line, Eq. (3), only those points should be take into account where the value of the plastic strain is higher than the established limit e0. This limitation results from the measuring accuracy of the total strain and from the operations of subtraction in Eq. (2). For steels, the value e0 = 0.01% is recommended [12]. Many problems related to the strain limit e0 have been discussed in [10]. Linearization of the elastic part of Eq. (1) can be done in the same manner 0 r Y ¼ log f þ bX ; ð4Þ E where Y = log (ea,e), and X = log (2Nf). Ramberg and Osgood [1,13] proposed a method of describing stresses and strains in materials subjected to constant amplitude cyclic loading. This method is widely applied in fatigue analyzes as an important source of information about the material behaviour [5,7]. Like in the

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

a

b

σ

1969

σ

εa,p,m = εa,t – εa,e

εa,p,m = εa,t – εa,e

ε

εa,p,m

ε

εa,e

εa,p,m

εa,e

Fig. 2. Calculated (a) and measured (b) plastic parts of the strain amplitude.

Manson–Coﬃn–Basquin equation, the total strain amplitude is divided into elastic and plastic components, but described as a function of stress amplitude ra and not fatigue life Nf ra ra n10 ea;t ¼ ea;e þ ea;p ¼ þ : ð5Þ E K0 The ﬁrst term of the equation is linear (the elastic component), and the second term is logarithmic (the plastic component). The stress–strain curve is usually presented in a linear coordinate system, Fig. 3a, but in the double-logarithmic coordinate system, the terms of the equations are expressed by straight lines, Fig. 3b. Thus, in order to determine the material parameters K0 and n0 , the plastic part of the Eq. (5) must be linearized. Linearization is performed by ﬁnding the logarithms of the equation 0

ra ¼ K 0 ena;p ;

ð6Þ

and thus obtaining

Y ¼ logðK 0 Þ þ n0 X ;

ð7Þ

where Y = log (ra), and X = log (ea,p). The experimental points are obtained from the constant amplitude fatigue tests and ﬁtted using the least-squares method. This approximation only takes those points into account where the plastic strain is greater than the assumed limit value e0. Here, the test results used for determining the Manson–Coﬃn–Basquin curve are usually applied [9]. Since many materials stabilize after a relatively short time, it is possible to obtain the experimental points using a more eﬃcient method that uses only one specimen. These methods are: incremental step test, multiple steps with increasing strain, multiple steps with decreasing strain and a method using random cyclic strain amplitude tests [14,15]. The experimental results (pairs of amplitudes of total stress and strain) are considered in the same way as those obtained from constant amplitude tests for stabilised hysteresis, and are ﬁtted as described above. However, the results obtained

Fig. 3. Example of the Ramberg–Osgood curve according to Eq. (5): (a) linear scale; (b) logarithmic scale.

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for materials with unstable material behaviour can diﬀer extensively depending on the used method [9,14]. 2. The compatibility terms Eqs. (1) and (5) have similar structures in that both total strain amplitudes consist of the sum of plastic and elastic parts. By comparing the elastic components, the following equations can be derived RO eMC a;e ¼ ea;e ; r0f

ra ; E E logðr0f Þ þ logð2N f Þb ¼ logðra Þ b

ð2N f Þ ¼

ð8Þ

and for the plastic parts of Eqs. (1) and (5) RO eMC a;p ¼ ea;p ; r 10 a n ¼ e0f ð2N f Þc ; K0

ð9Þ

1 ¼ logðe0f Þ þ logð2N f Þc: n0 Introducing the log (ra) part from Eq. (8) into (9) leads to the following expression 1 logðr0f Þ þ logð2N f Þb logðK 0 Þ 0 ¼ logðe0f Þ þ logð2N f Þc n ð10aÞ 1 1 1 logðr0f Þ 0 þ logð2N f Þ b 0 c logðK 0 Þ 0 ¼ logðe0f Þ ð10bÞ n n n ½logðra Þ logðK 0 Þ

Material parameters as well as number of cycles appear in Eq. (10b). This equation shows three terms containing only material parameters which do not depend on the number of cycles. However, this equation should be valid for the entire range of cycles, which then leads to 1 1 logð2N f Þ b 0 c ¼ 0 if; and only if; b 0 c ¼ 0; ð11Þ |ﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄ} n n

assumed that the material is stable during fatigue tests and does not exhibit signiﬁcant softening or hardening eﬀects. Therefore, the constants E, K0 , n0 , e0f , r0f , b and c do not depend on the number of cycles Nf or on the values of strain ea or stress ra. This assumption may be invalid for some materials. 3. Problems when compatibility is not ensured The known methods for determining material constants do not ensure compatibility. The material constants are usually determined separately, and the sets of data points are treated as separate experimental results. If Eqs. (12) and (14) are not adhered to, a loss of equality between the plastic and elastic parts occurs in both formulations. To make the problem clear the ratios of the elastic, respectively of the plastic strain over the total stress amplitude were computed according to the ﬂow chart showed in Fig. 4 and presented in Fig. 5. When compatibility is given, the ratio of the elastic part of the strain and the ratio of the plastic part of the strain of the Ramberg–Osgood and the Manson–Coﬃn–Basquin equations is always one for all total strain amplitudes, and therefore for whole range of cycles. Lack of compatibility causes the ratios to deviate from one and yields to non-linear curves. This is incongruent because each tested specimen possesses elastic and plastic strain parts independent of whether it is used for the analysis of the strain–life curve or the stress–strain curve [1,5]. The component terms should be equal in both formulations. When compatibility is not given the line of the ratio of the elastic part deviates from one to higher amplitudes, and the line of the ratio of the plastic part shows large deviations to smaller amplitudes.

6¼0

b n0 ¼ ¼ n0comp: c Substitution of Eq. (12) into (9) leads to 1 1 logðr0f Þ 0 logðK 0 Þ 0 ¼ logðe0f Þ n n r0f 0n0 ¼ ef K0 K0 ¼

ð12Þ

ð13Þ

r0f

0 ð14Þ b ¼ K comp: ðe0f Þc Eqs. (12) and (14) are the so-called compatibility equations and they couple the coeﬃcients and exponents which appear in the Ramberg–Osgood and Manson–Coﬃn–Basquin formulations. This yields the possibility of determining the coeﬃcient and exponent in the stress–strain curve (5) directly from the strain–life curve (1). In Eq. (10) it is assumed, that the number of cycles to failure does not inﬂuence the elastic–plastic properties of the material. Thus, it is

Fig. 4. The ﬂow chart for computing the numerical quantities used in Figs. 5 and 6.

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

1971

Fig. 5. Computed ratios for elastic and plastic strains by neglecting the compatibility.

Thus, any of the following applications connecting the parts of Eqs. (1) and (5) in further calculations can lead to large deviations in the fatigue life of a component. This can be shown if the number of cycles to the fatigue crack occurrence is determined, either by using Eq. (1), or by using the stress amplitude and the following equation: 1 1 ra b Nf ¼ : ð15Þ 2 r0f

If compatibility is satisﬁed Eqs. (15) and (1) leads to the same number of cycles. The consecutive computational steps are shown in Fig. 4 and the resulted values on Fig. 6. The numbers of cycles Nf and N MCB obtained by f way of the two calculations methods are diﬀerent if the compatibility equations are not satisﬁed. That means that the equation leads to two diﬀerent stress amplitudes for one number of cycles or it leads to two diﬀerent number of cycles for one stress amplitude, see Fig. 6. Moreover,

Fig. 6. Calculated number of cycles by neglecting the compatibility.

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A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

the numerically determined stress–life characteristic MCB ðrRO Þ does not produce a straight line in the doua Nf ble-logarithmic coordinate system, thus disagreeing with Eq. (15) used for its description. This can cause signiﬁcant deviation in the determination of the cycle numbers, when the ﬁnal formulae for determining fatigue life are derived on the assumption that the elastic and plastic parts in Eqs. (1) and (5) are equal.

z

z1 P1 ( x1 , y1 , z1 )

R (l , m , n )

4. The proposed method From Eqs. (1) and (5) it appears, that one value of the total strain amplitude, ea,t, corresponds to only one stress amplitude, ra, and to a single number of cycles, Nf. In such a case, it is possible to formulate the total characteristics expressed by the curve (ea,t–ra–Nf) in a three-dimensional space. Fig. 7a shows such characteristics also depicting curves for the parts of elastic strain (ea,e–ra–Nf) and plastic strain (ea,p–ra–Nf). By a projection of the three-dimensional characteristics, the following curves can be obtained: projection to the plane (ea,t–ra) gives the trajectory of the cyclic stress–strain curve (Ramberg–Osgood equation), projection to the plane (ea,t–Nf) gives the strain–life curve (Manson–Coﬃn–Basquin equation), and projection to the plane (ra–Nf) gives the stress–life curve, expressed with Eq. (15). Particular projections for 3-D characteristics are shown in Fig. 7b–d.

y1

y

x1 x Fig. 8. Points in the three-dimensional space with the regression line determined by point P1 and directional vector R.

In this new method, all six cyclic characteristics are determined through the approximation of the regression line in the 3-D space, and there is no dependence on the regression direction. The proposed method retains compatibility. Location of the straight line (ea,p–ra–Nf) in 3-D space can be deﬁned by coordinates of the point P1(x1, y1, z1) located on this straight line and the directional vector R(l, m, n) to which the straight line is parallel [15], Fig. 8.

Fig. 7. (a) Three-dimensional space with the strain–stress–life curve (ea–ra–Nf) and its projections on (b) strain–stress (ea–ra) plane, (c) stress–life (ra–Nf) plane, and (d) strain–life (ea–Nf) plane.

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

It is proposed to approximate the points with the least-square method, minimizing the sum of the squares of the distances of the points from the straight line. There are many eﬃcient methods for such regressions in the 3-D space [16,17]. The location of the straight line in 3-D space also determines the positions of its projections on the particular planes. Thus, the material parameters of interest can be determined directly from the following equations: m ; l 0 K 0 ¼ 10ðy 1 x1 n Þ ;

n0 ¼

l m c¼ ; b¼ ; n n e0f ¼ 10ðx1 z1 cÞ ; r0f ¼ 10ðy 1 z1 bÞ :

ð16Þ

These equations have been derived according to the following assumptions: x ¼ logðea;p Þ;

y ¼ logðra Þ;

z ¼ logðN f Þ:

ð17Þ

As in the conventional method, the established limit e0 is considered in the 3-D method. While ﬁtting the line, only those points were considered where the value of plastic strain is higher than e0. However, it is not recommended to do this directly since omission of such points can greatly inﬂuence the stress–life curve (ra–Nf), where the elastic part of the strain plays a signiﬁcant role. Therefore, it is proposed to recalculate the plastic strain parts with e0 < 0.01% so that the new values do not change the strain–life curve (ea–Nf). These new values then lie directly on the line of the plastic part of the Ramberg–Osgood equation, which was determined without considering these points. However, the stress values which form the third plane of these tests are not corrected, and are subject to further scatter. Subsequently, a second regression is carried

1973

out in order to recalculate the location of the (ea,p–ra–Nf) line. The algorithm of the computations is shown in Fig. 9. 5. Practical implementation The proposed method was used for calculating the fatigue parameters from Ramberg–Osgood and Manson–Cofﬁn–Basquin equations for spray-compacted aluminium alloys (DISPAL = DISPersion hardened ALuminium) made by PEAK Werkstoﬀ GmbH. These materials are characterized by their high Young’s modulus, good wear resistance and a low coeﬃcient of thermal expansion. In the framework of a bilateral project, several tests on diﬀerent materials were carried out by varying the test conditions. A detailed description of the materials and the static and fatigue tests can be found in previous studies [18,19]. For each material, the constant strain amplitude tests were performed under alternating load and the numbers of cycles to crack initiation were recorded. The stress amplitude was obtained from the recorded stable hysteresis loop at half of the number of cycles to crack initiation. For comparison purposes, the conventional method of determining the fatigue parameters [18,19] was also calculated according to the algorithm shown in Fig. 7. The obtained results can be found in Tables 1 and 2. Since the conventional method does not satisfy the compatibility relations, two simple parameters, V K 0 and V n0 , were deﬁned V K0 ¼

K0 K 0comp:

;

V n0 ¼

n0 n0comp:

:

ð18Þ

These parameters show the diﬀerences between material constants K0 and n0 obtained from Eqs. (12),(14), and

Fig. 9. Algorithms for calculation of the material constants K0 , n0 ; e0f , c, r0f and b.

1974

Number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 21 22 23 24

Material and heat treatment

Pressing

S225 F S250 F S220 F S260 F S250 F S250 F S250 F S226 T6 S250 F S250 F S250 F S230B T6 S232 T6 S232 T6 S691 T6 S693 T6 S691 T6 S693 T6

19 16 13 7A 15 29 34 5B 34 28 30 6A 2 2 7 40 7 40A

Raw part diameter (mm)

Position in the raw part

Temperature (°C)

Young’s modulus E (GPa)

Cyclic values according to Manson–Coﬃn–Basquin curve Fatigue strength coeﬃcient (MPa) r0f Conv. 3-D

Fatigue strength exponent b Conv. 3-D

Fatigue ductility coeﬃcient (m/m) e0f Conv. 3-D

Fatigue ductility exponent c Conv. 3-D

Fitting quality (%) R2 Conv. 3-D

30 30 30 30 30 60 100 30 100 60 60 30 30 30 30 30 30 30

l l l l l l l l c l l l l l l l l l

20 20 20 20 200 20 20 20 20 20 20 20 20 200 20 20 200 200

100.0 100.0 92.5 90.0 90.0 100.0 100.0 92.5 100.0 97.5 97.5 92.5 92.5 86.0 75.0 73.0 73.0 71.0

241.4 394.0 228.1 420.6 284.0 455.0 330.3 499.5 369.2 407.9 370.5 411.4 654.9 450.0 1030.5 866.6 499.7 464.6

0.063 0.099 0.082 0.061 0.073 0.110 0.075 0.063 0.082 0.100 0.088 0.063 0.066 0.086 0.115 0.098 0.082 0.077

0.046 0.035 0.121 0.100 0.038 0.048 0.038 0.019 0.029 0.049 0.035 0.224 0.069 1.219 10.054 15.781 1.358 0.573

0.431 0.359 0.500 0.620 0.324 0.374 0.356 0.484 0.333 0.369 0.330 0.668 0.739 0.847 1.237 1.445 0.851 0.765

97.6 93.8 96.6 98.4 98.3 98.6 94.6 97.1 99.2 98.9 97.7 86.7 97.3 96.8 99.3 97.8 97.1 98.6

241.6 396.9 228.7 420.7 284.1 455.5 331.6 499.9 369.5 408.3 371.3 413.4 655.1 449.8 1030.6 867.4 499.8 463.9

0.063 0.099 0.082 0.061 0.073 0.110 0.075 0.063 0.082 0.100 0.088 0.063 0.066 0.086 0.115 0.099 0.082 0.076

0.046 0.036 0.124 0.107 0.038 0.048 0.039 0.019 0.029 0.049 0.035 0.244 0.072 1.482 10.945 36.385 1.504 0.605

0.432 0.361 0.502 0.628 0.324 0.374 0.358 0.487 0.333 0.369 0.331 0.678 0.744 0.868 1.247 1.557 0.862 0.772

97.5 93.8 96.4 98.3 98.4 98.6 94.8 97.0 99.2 98.9 97.7 85.4 97.3 96.2 99.2 97.5 96.9 98.6

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

Table 1 Cyclic values according to the Manson–Coﬃn–Basquin curve

Number

1 2 3 4 5 6 7 8 9 10 11 12 13 14 21 22 23 24

Material and heat treatment

S225 F S250 F S220 F S260 F S250 F S250 F S250 F S226 T6 S250 F S250 F S250 F S230B T6 S232 T6 S232 T6 S691 T6 S693 T6 S691 T6 S693 T6

Pressing

19 16 13 7A 15 29 34 5B 34 28 30 6A 2 2 7 40 7 40A

Raw part diameter (mm)

30 30 30 30 30 60 100 30 100 60 60 30 30 30 30 30 30 30

Position in the raw part

l l l l l l l l c l l l l l l l l l

Temperature (°C)

20 20 20 20 200 20 20 20 20 20 20 20 20 200 20 20 200 200

Young’s modulus E (GPa)

100.0 100.0 92.5 90.0 90.0 100.0 100.0 92.5 100.0 97.5 97.5 92.5 92.5 86.0 75.0 73.0 73.0 71.0

Cyclic values according to Ramberg–Osgood curve Cyclic hardening coeﬃcient (MPa) K0 Conv. 3-D K 0Comp: 393.7 1009.3 380.2 465.3 590.4 1099.0 649.1 730.4 877.0 923.7 912.2 464.1 956.3 248.6 675.3 572.5 273.3 333.7

379.4 991.4 322.2 523.4 593.8 1108.5 654.0 835.9 879.9 924.9 902.3 471.6 828.1 432.5 826.3 691.0 480.9 487.6

380.0 988.5 322.7 528.4 594.1 1109.8 654.5 839.5 880.1 925.0 901.0 473.6 832.0 441.0 831.2 718.1 485.2 491.3

V K0

Cyclic hardening exponent n0 Conv. 3-D n0Comp:

V n0

Fitting quality (%) R2 Conv. 3-D Comp.

1.036 1.021 1.178 0.881 0.994 0.990 0.992 0.870 0.997 0.999 1.012 0.980 1.149 0.564 0.812 0.797 0.563 0.679

0.152 0.277 0.188 0.082 0.225 0.292 0.208 0.112 0.245 0.271 0.267 0.091 0.105 0.025 0.066 0.041 0.016 0.047

1.036 1.011 1.144 0.832 0.996 0.995 0.995 0.857 0.998 0.999 1.007 0.966 1.176 0.248 0.709 0.597 0.170 0.474

95.5 99.2 99.3 99.6 99.6 99.7 97.9 99.8 98.7 99.7 99.7 96.9 99.9 98.5 99.9 99.9 99.1 96.0

0.147 0.275 0.164 0.098 0.226 0.293 0.209 0.130 0.246 0.271 0.266 0.093 0.089 0.099 0.092 0.063 0.095 0.099

0.147 0.274 0.164 0.099 0.226 0.293 0.209 0.130 0.246 0.271 0.266 0.094 0.089 0.102 0.093 0.068 0.096 0.100

95.4 99.2 99.0 99.5 99.6 99.7 97.9 99.7 98.7 99.7 99.6 96.9 99.7 97.2 99.7 99.7 97.3 94.9

95.4 99.2 99.0 99.4 99.6 99.7 97.9 99.7 98.7 99.7 99.6 96.9 99.7 97.1 99.7 99.7 97.3 94.9

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

Table 2 Cyclic values of the Ramberg–Osgood curve

1975

1976

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

Fig. 10. Stress–strain curves obtained with the conventional and the 3-D method.

Fig. 11. Strain–life curves obtained with the conventional and the 3-D method.

(16). The stress–strain and strain–life curves for three selected materials are shown in Figs. 10 and 11. This new method of evaluation will now also be used in the context of the Collaborative Research Centre CRC 666 project ‘‘Integral Sheet Metal Design with Higher Order Bifurcation” (started 2005). The aim of two subprojects in the CRC 666 at the chair of System Reliability and Machine Acoustic SzM of the Technical University of Darmstadt is to analyze and evaluate material characteristics resulting from the new manufacturing method called

‘‘linear ﬂow splitting” [20,21]. With this new processing method, integral design proﬁles are produced by utilising very high degrees of deformation and strong surface hardening eﬀects. These proﬁles are aﬄicted with gradients (residual stress, hardness and microstructure) that require exact knowledge of the local conditions. Here, it was important to closely analyze the compatibility conditions and the methods for evaluating the strain–life curve and the stress–strain curve such that diﬀerences in the stress– strain and the strain–life curve are coming from diﬀerent

A. Niesłony et al. / International Journal of Fatigue 30 (2008) 1967–1977

existing local condition and not from the evaluation method. 6. Conclusion The proposed 3-D method uses a spatial approximation of the test results in order to determine the cyclic parameters. The method yields complete compatibility. The straight line is determined by the least-squares analysis in the 3-D space and the accuracy of the regression is indicated. The conventional methods depended on the regression line (3), (4), or (7), and on the direction (x-axis or y-axis) in which the regression was carried out. This dependence had a large inﬂuence on the cyclic coeﬃcients and exponents, and compatibility was not always given because the assumptions did not allow such dependences. In contrast, the new 3-D analysis method guarantees compatibility. However, for certain materials, it cannot describe the stress–strain curve adequately. For various materials, the determined stress–strain curve derived from the compatibility condition does not agree with the test results, and therefore the materials do not possess compatibility behaviour. Describing this material behaviour is one object of further investigation in the framework of the CRC 666 and a method will be presented soon. Acknowledgements The introduced results were developed in the framework of the German research foundation in the subproject C2 of the CRC 666, entitled ‘‘Developing Methods for the Estimation of the structural durability and System Reliability of Integrated Construction of Metal Sheets”. Furthermore, the authors thank the PEAK Material GmbH in Velbert, Germany, for the provision of material data through which the applicability of this analysis method could be shown. Dr. Adam Nieslony is pleasured to acknowledge the Alexander von Humboldt Foundation for sponsoring his stay in Germany. References [1] Mitchell MR. Fundamentals of modern fatigue analysis for design. In: Lampman Steven R, editor. ASM handbook. Materials Park: ASM International; 1996. p. 229–49.

1977

[2] Manson SS. Fatigue: a complex subject – some simple approximation. Exp Mech 1965;5:193–226. [3] Coﬃn LF. A study of the eﬀect of cyclic thermal stresses on a ductile metal. Trans ASME 1954;76:931–50. [4] Basquin OH. The exponential law of endurance tests. Am Soc Test Mater Proc 1910;10:625–30. [5] Rice RC, Leis BN, Berns HD, Nelson DV, Lingenﬂeser D, Mitchell MR. Fatigue design handbook, Warrendale: SAE; 1988. p. 369. [6] Plumtree A, Abdel-Raouf HA. Cyclic stress–strain response and substructure. Int J Fatigue 2001;23:799–805. [7] Skelton RP, Maier HJ, Christ H-J. The Bauschinger eﬀect, Masing model and the Ramberg–Osgood relation for cyclic deformation in metals. Mater Sci Eng A 1997;238:377–90. [8] ASTM Standard E606-92: Standard practice for strain-controlled fatigue testing. In: Annual book of ASTM standards, vol. 03.01. ASTM; 1997, p. 523–37. [9] Ba¨umel A, Seeger T. Material data for cyclic loading. Mater Sci Monogr 1990;61(Suppl. 1). Amsterdam: Elsevier Science Publishers. [10] Williams CR, Lee Y-L, Rilly JT. A practical method for statistical analysis of strain–life fatigue data. Int J Fatigue 2003;25: 427–436. [11] Williams CR, Lee Y-L, Rilly JT. Characterizing statistical strain–life fatigue parameters. Int J Mater Prod Technol 2004;20(Part 1/3): 220–30. [12] Testing and Documentation Guideline for the Experimental Determination of Mechanical Properties of Steel Sheets for CAE-Calculation, STAHL-EISEN-Pru¨fbla¨tter (SEP) des Stahlinstituts VDEh, SEP 1240, 1st ed. Verein Deutscher Eisenhu¨ttenleute VDEh, Du¨sseldorf, 2006; p. 64 [in German]. [13] Ramberg W, Osgood WR. Description of stress–strain curves by three parameters, Technical Note No. 902, National Advisory Committee for Aeronautics, Washington DC, 1943. [14] Jones A, Hudd RC. Cyclic stress–strain curves generated from random cyclic strain amplitude tests. Int J Fatigue 1999;21: 521–30. [15] Landgraf RW, Morrow J, Endo T. Determination of the cyclic stress– strain curve. J Mater JMLSA 1969;4(1):176–88. [16] Bronstein IN, Semendjajew KA. Handbook of mathematics. Berlin: Springer; 2004. p. 1157. [17] Matlab user’s guide, MathWorks Inc., Version 7, 2006, p. 311. [18] El Dsoki C, Krug P, Kaufmann H. Spray compacted high performance aluminium alloys, Konstruktion, vol. 10. VDI-Verlag; 2006. p. 13–14 [in German]. [19] El Dsoki C, Landersheim V, Kaufmann H, Bruder T, Krug P, Bohn T, et al. The fatigue behaviour of spray compacted aluminium materials. Material properties for component design data: from specimen to component. Materialwiss. Werkstoﬀtech. 2006;37(12): 1006–17. [20] Groche P, Vucic D. Multi-chambered proﬁles made from highstrength sheets. In: Production engineering, vol. XIII/1. Annals of the WGP, Hannover; 2006. p. 67–70. [21] Groche P, Vucic D, Jo¨ckel M. Basics of linear ﬂow splitting. J Mater Process Technol 2007;183(2–3):249–55.

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