Synthetic S–N curves for sintered steels based on density,hardness,and local stress parameters: A data analysis

Abstract

While components produced by melt metallurgy have long been subject to established rule sets for proof of strength, a comparable guideline has now been developed for press-and-sinter powder metallurgical components, taking into account their characteristic residual porosity. This paper presents a method that enables the generation of a synthetic S–N curve using hardness, density, and local stress parameters as input parameters. The development of this method required a comprehensive literature review, gathering data on fatigue strength parameters in relation to density, hardness, as well as specimen geometry and load information. The compiled data was utilized to deduce relationships and formulate an algorithm capable of estimating fatigue strength associated with a 50% failure probability. Moreover, the method can supply information regarding the S–N curve’s trajectory within the finite life region. Coupled with results on statistical scatter derived from the literature, comprehensive synthetic S–N curves for sintered steels can be constructed.

Keywords

synthetical S–N curve sintered steels fatigue limit data analysis strength assessment

Introduction

Although the Forschungskuratorium Maschinenbau (FKM) guideline “Analytical Strength Assessment”¹ provides a well-established methodology for components made by melt metallurgy, a comparable approach for sintered steels has only recently been introduced. The new guideline “Analytical Strength Assessment of Components Made of Sintered Steels”² now enables a computational proof of strength for press-and-sinter components, despite their residual porosity. This paper aims to present and detail the computational procedure proposed in the guideline for assessing the fatigue strength of components made from sintered steels. It should be noted that some of the values and estimations used in the strength assessment may differ from determined properties in this paper, as they are conservatively adjusted to ensure safe component design.

The concept for the computational proof of strength under cyclic loading of porous sintered steel components must incorporate the following influencing factors:

density,

size, geometry, and type of load on the specimen or component,

static mean stresses,

alloy effects, and

heat treatments.

The variable density of sintered steels, in particular, presents a unique characteristic that distinguishes sintered steels from the materials made by melt metallurgy featured in common guidelines. Recent works^3–6 have presented new approaches for selected steels, which are taken up and extended here to be generally used for a wide range of sintered steels.

A crucial component of the strength proof involves determining a synthetic or a design S–N curve, considering the factors mentioned above. Therefore, this paper aims to describe how a database of mechanical properties of sintered steels was constructed based on a literature survey, which was then used to develop approaches for determining synthetic S–N curves for sintered steels.

Materials and methods

The following will detail the approach for determining the relationships necessary for the creation of a synthetic S–N curve. First, the process of collecting and processing empirical data is described. Subsequently, the individual influencing variables mentioned earlier are considered to determine the fatigue limit for a probability of failure of $P = 50 %$ . To generate a complete synthetic S–N curve, finally, details for determining the slope $k$ , the knee point $N_{k}$ , as well as for determining failure probabilities of $P \neq 50 %$ are given. A fundamental component for the development of design methods is the availability of empirical data, that is, results from fatigue strength tests and tensile tests. To develop a comprehensive concept for strength proof, which covers a wide range of the above-mentioned properties, fatigue strength characteristics of sintered steels were therefore collected from an extensive literature review and structured together in a database.

Structure of the database

The selection of materials, for which fatigue strength was to be extracted from the literature, was guided by the availability of data and the relevance of the material in the present market. For instance, there are many details on the fatigue strength of Fe–Cu–Ni steels without carbon from older works. This group of materials is hardly used today and was therefore omitted here. Conversely, there are steels demanded by the market for certain applications, but for which the data situation is so sparse that no correlations could be made visible. In such cases, inclusion was omitted. From these considerations, 11 material groups emerged:

Fe–C

Fe–Cu

Fe–Cu–C

Fe–1.75% Ni–1.5% Cu–0.5% Mo (–C)

Fe–4% Ni–1.5% Cu–0.5% Mo (–C)

0.85% Mo (pre-alloyed)–AE–C; AE: Cu, Ni a. o.

1.5% Mo (pre-alloyed)–2% Cu (–C)

1.5% Mo (pre-alloyed)–4% Ni–2% Cu (–C)

(1.5–1.8)% Cr (pre-alloyed)

(–0.2% Mo (pre-alloyed))–C

3% Cr (pre-alloyed)–0.5% Mo (pre-alloyed)–C

The compiled data comes from a variety of publications (dissertations, conference contributions, final reports, publications in technical journals, or internal investigations by institutes), some of which were published several decades ago.^3–168

The individual extracted data sets contain among others:

The fatigue endurance limit $S_{A}$ (as the target property of the data set)

The alloy composition

The density of the specimens

Information on heat treatment

Information on hardness

The specimen shape and type of load used to determine fatigue strength

Whenever possible, the raw data from individual S–N tests (nominal stress

S_{a}

and number of cycles to failure

N

) were extracted from the sources. Graphical representations of S–N curves with data points were digitized. Depending on the size of the original, the digitization may be associated with errors of up to

\pm 2 %

. This raw data was then uniformly evaluated to determine the fatigue limit at

S_{A}

, as well as the characteristics

k

and

C

(respectively

N_{k}

) of the S–N curve.

As in the FKM guideline¹ and DIN EN ISO 50100¹⁶⁹ the description of the S–N curve is based on the Basquin equation,

N = C {S_{a}}^{- k},

(1)

where

N

is the number of cycles to failure;

C

is the X-intercept at

S_{a} = 1

MPa;

S_{a}

is the nominal stress amplitude; and

k

is the slope of the fatigue strength curve in double-logarithmic coordinates; and only nominal stresses were calculated in the double-logarithmic system.

The raw data was evaluated using the method of weighted horizons and linear regression, as it is extremely flexible and includes many other evaluation methods as well as some other rarely used procedures. Another advantage is that the evaluation method in the experimental test does not require strict adherence to an order and does not demand a constant linear or logarithmic step change.

If the raw data was not available in the source, the value for $S_{A}$ stated in the source was used, and the underlying evaluation method (such as staircase method, maximum likelihood method, horizon method, evaluation according to Hück or Dixon and Mood) was indicated. Thus, the statistical evaluation procedures underlying the calculated values for $S_{A}$ differ between the individual data sets. Since a large number of sources were evaluated, the number of specimens used for each S–N curve also varies greatly. S–N curves in which only five or fewer specimens were used were excluded from the evaluation and not incorporated into the database.

To obtain characteristic values for local stress parameter depending on specimen size, geometry, and load type, the shape factor $K_{t}$ and the highly loaded volume $V_{90}$ were determined in a finite-element analysis for each sample geometry in connection with the load type. The highly loaded volume $V_{90}$ was defined as the volume in the specimen in which the first principal normal stress $σ_{I}$ is at least 90% of the greatest principal normal stress. The shape factor is defined as the greatest principal normal stress divided by nominal maximal stress $S$ :

K_{t} = \frac{σ_{I, max}}{S}

(2)

The type of load was differentiated between:

axial load

90°-bending

0°-bending

rotational bending

three-point bending

torsion

Figure 1 shows an example of a test specimen frequently used in literature with a notch radius of $r = 0.25$ mm. The corresponding values for $K_{t}$ and $V_{90}$ are shown in Table 1.

Figure 1.

Example of a specimen geometry from the database with notch radius $r = 0.25$ mm.

Table 1.

Example for FEA results for specimen geometry shown in Figure 1.

Load type	Thickness	$K_{t}$	$V_{90}$ ( $R = 0$ )	$V_{90}$ ( $R = - 1$ )
	(mm)	(–)	(mm $^{3}$ )	(mm $^{3}$ )
Axial	4	4.44	1.20 $\times 10$ ⁻²	1.20 $\times 10$ ⁻²
	5	4.43	1.55 $\times 10$ ⁻²	1.55 $\times 10$ ⁻²
	7	4.43	2.14 $\times 10$ ⁻²	2.14 $\times 10$ ⁻²
90°-bending	4	3.17	5.66 $\times 10$ ⁻³	1.13 $\times 10$ ⁻²
	5	3.15	7.29 $\times 10$ ⁻³	1.46 $\times 10$ ⁻²
	7	3.15	1.04 $\times 10$ ⁻²	2.09 $\times 10$ ⁻²
0°-bending	4	3.29	8.78 $\times 10$ ⁻⁴	1.76 $\times 10$ ⁻³
	5	3.44	1.07 $\times 10$ ⁻³	2.41 $\times 10$ ⁻³
	7	3.62	1.45 $\times 10$ ⁻³	2.90 $\times 10$ ⁻³
Torsion	4	2.73	9.20 $\times 10$ ⁻⁴	1.84 $\times 10$ ⁻³
	5	2.79	1.08 $\times 10$ ⁻³	2.15 $\times 10$ ⁻³
	6	2.60	1.22 $\times 10$ ⁻³	2.45 $\times 10$ ⁻³
	7	2.41	1.34 $\times 10$ ⁻³	2.68 $\times 10$ ⁻³

FEA: finite-element analysis.

Evaluation

Density and hardness

With conventional pore-free steels, there is a statistically well-documented proportionality between axial fatigue strengths or bending fatigue strengths and tensile strengths up to 1200 MPa.^170–172 Therefore, the easily measurable tensile strength of this group of materials is well suited to estimate bending or axial fatigue strengths, as is applied in the FKM guideline¹ for the material groups represented there. This option is only possible with limitations for porous sintered steels: up to carbon contents of about 0.5–0.6%, bending and tensile fully reversed fatigue strengths also increase proportionally with tensile strength. However, with higher carbon contents, tensile strength decreases while hardness and fatigue strengths continue to increase.^26,173–177 Therefore, it is more purposeful for sintered steels to use hardness in connection with density to estimate fatigue strength. To derive a correlation, all data sets were used in which the hardness and density are specified and whose highly stressed volume is approximately $V_{90} = 10$ mm $^{3}$ , at a stress ratio of $R = - 1$ . Only with a similarly large $V_{90}$ a comparability of fatigue strengths is possible. A $V_{90} = 10$ mm $^{3}$ was chosen because the majority of data is available for this size. The vast majority of fatigue strengths were measured under flat bending with fully reversed stress with an unnotched specimen according to DIN ISO EN 3928.¹⁷⁸ In the USA, an unnotched specimen according to MPIF standard 56¹⁷⁹ is of great importance, which is tested under rotational bending and under this stress has a similarly large $V_{90}$ .

The Balshin equation can be used as an approach to describe the fatigue strength in relation to hardness. With it, density dependencies can be quantitatively described.

P = P_{0} {(\frac{ρ}{ρ_{0}})}^{m}

(3)

where

P

is the property;

ρ

is the density;

m

is thedensity exponent; and index 0 is the non-porous state.

S_{A} = S_{A 0} {(\frac{ρ}{ρ_{0}})}^{m_{ρ, S}}

(4)

and

H = H_{0} {(\frac{ρ}{ρ_{0}})}^{m_{ρ, H}}

(5)

can be formulated for fatigue limit

S_{A}

and hardness

H

. According to Beiss,¹⁴⁷ an average density exponent of

m_{ρ, S} = 5.0

was found from 40 density dependencies of bending or fatigue strengths of sintered steels. Likewise, more than 100 density dependencies of the Vickers hardness of sintered steels from water-atomized powders were evaluated according to equation (5), resulting in an average density exponent of

m_{ρ, H} = 4.35

.¹⁴⁷ Dividing the equations by each other results in:

S_{A} = H (\frac{S_{A 0}}{H_{0}}) {(\frac{ρ}{ρ_{0}})}^{m_{ρ, σ} - m_{ρ, H}} .

(6)

The exponent in equation (6) can therefore be set on average at

m_{ρ, σ} - m_{ρ, H} = 0.65

. The industry-standard densities of sintered steels are 7.0 g/cm

^{3}

. Therefore, it makes sense to formulate equation (6) not for full density but for 7.0 g/cm

^{3}

. The result is:

S_{A} = H (\frac{S_{A 7.0}}{H_{7.0}}) {(\frac{ρ}{ρ_{0}})}^{0.65} .

(7)

The Vickers hardness at 7.0 g/cm³ is considered process- and material-specific and can be determined in simple tests for the most varied industrial process variants. To be able to determine

S_{A 7.0} / H_{7.0}

depending on hardness as a basic measurement variable, all fatigue strength values for the hardness either in Vickers or in a scale convertible to Vickers were used from the database. For this purpose, own conversion rules were partly determined to be able to convert the individual hardness scales into Vickers hardness values. Figure 2 plots the quotient

S_{A 7.0} / H_{7.0}

of the available data over

H_{7.0}

Figure 2.

$S_{A 7.0} / H_{7.0}$ as a function of Vickers hardness at $ρ = 7.0$ g/cm $^{3}$ .

The data can be well described with a second-degree polynomial:

S_{A 7.0} / H_{7.0} = - 2.247 + 5.822 \cdot H_{7.0}^{- 0.1091}

(8)

Given the hardness at 7.0 g/cm

^{3}

it is now possible to estimate the fatigue strength of a material with equations (7) and (8) for a

V_{90} \approx 10

^{3}

for any hardness.

Estimation of the fatigue strength for arbitrary specimen geometries and load cases

The size and notch effect for sintered steels can be well described with the concept of highly stressed volume, as demonstrated in the literature.^4,5,133,152 The relationship between the tolerable stress amplitude and the highly stressed volume can be described using a Weibull approach.

P_{S V} = \exp [- \frac{V}{V_{ref}} {(\frac{σ - σ_{0}}{σ_{ref}})}^{n}]

(9)

where

σ_{0}

is the stress limit for crack growth;

σ_{ref}

is the reference stress at

V = V_{ref}

reference volume; and

n

is the Weibull exponent.

As an example, Figure 3 shows the fatigue limit as local stress amplitude $σ_{A}$ as a function of $V_{90}$ for a carbonitrided Fe–Cu–C steel. All values for $σ_{A}$ are normalized to $ρ = 7.0$ g/cm $^{3}$ with equation (4). To obtain a universally applicable approach to the equation depending on the previously determined fatigue strength for the unnotched specimen with $V_{90} \approx 10$ mm $^{3}$ , a normalized approach must be found. For this purpose, all data sets were used in which fatigue strengths for the same materials were measured on specimens with different shaped notches. In the literature, there are hardly any complete data sets in which fatigue strength has been systematically measured across several orders of magnitude of $V_{90}$ . Nevertheless, a consistent picture with relatively few outliers can be drawn for the normalized progression of fatigue strength depending on the highly stressed volume. Figure 4 shows the fatigue strength normalized to the fatigue strength at a density of 7.0 g/cm $^{3}$ as a function of the highly stressed volume. The available data sets come from dissertations and research projects.^5,152,180 Only here relationships for selected sintered steels were systematically investigated, and specimens with $V_{90}$ between 0.001 and 1000 mm $^{3}$ were systematically tested. Figure 4 shows the normalized data. For the context depicted in Figure 4, the following expression can be formulated with good approximation:

σ_{A} = σ_{A} (V_{90} \approx 10 {mm}^{3}) (0.86 + 0.2706 \cdot V_{90}^{- 1 / 3.635}) .

(10)

Figure 3.

$σ_{A}$ at $ρ = 7.0$ g/cm $^{3}$ and $R = - 1$ as a function of $V_{90}$ for a carbonitrided Fe–Cu–C steel.

Figure 4.

$σ_{A 7.0}$ normalized to $σ_{A 7.0}$ at $V_{90} \approx$ 10 mm $^{3}$ as a function of $V_{90}$ .

Influence of mean stress: Haigh diagram

As shown by our own investigations in Beiss et al.,¹⁶⁷ the Haigh diagram for sintered steels can only be established on the basis of $σ_{A}$ at $R = - 1$ . Therefore the following equation

\frac{σ_{A} (R = 0)}{σ_{A} (R = - 1)} = \frac{1}{2} + \frac{1}{2} [1 - \exp (- \frac{140 M P a}{σ_{A} (R = - 1)})]

(11)

was established to calculate

σ_{A} (R = 0)

as a function of

σ_{A} (R = - 1)

. Then mean stress sensitivity

M

according to FKM guideline¹ can be established as

M = \frac{\exp [- \frac{140 MPa}{σ_{A} (R = - 1)}]}{2 - \exp [- \frac{140 MPa}{σ_{A} (R = - 1)}]} .

(12)

If only positive mean stress values are considered, the fatigue limit

σ_{A}

becomes

σ_{A} (- 1 \leq R \leq 0) = σ_{A} (R = - 1) / (1 + \frac{1 + R}{1 - R} M)

(13)

in the interval $- 1 \leq R \leq 0$ . For the interval $0 \leq R \leq 0.5$

σ_{A} (0 \leq R \leq 0.5) = σ_{A} (R = 0) \cdot \frac{1.75 + M}{1.75 + ((1 + R) / (1 - R)) M}

(14)

was proposed. For $0.5 \leq R \leq 0.8$

σ_{A} (0.5 \leq R \leq 0.8) = σ_{A} (R = 0.5) \cdot \frac{4 + 3 M}{4 + ((1 + R) / (1 - R)) M} .

(15)

However, there were just a few experimental data points to backup equation (15). For

R \geq 0.8

there is insufficient amount of data in literature so it cannot be claimed that mean stress sensitivity can be set to 0. Also for

R < - 1

there is not sufficient data as the largest proportion of specimens was tested under bending load.

Finite life regime

As slope $k$ and knee point $N_{k}$ is given or was recalculated for each data set, a statistically well validated mean value for both $k$ and $N_{k}$ can be calculated. Given the variance in the quality of S–N data from each curve, filter criteria were introduced to exclude poor data from the evaluation:

At least 10 fractured specimens with $N < N_{k}$

$N_{k} / N_{min} > 10$ ( $N_{min}$ : minimum number of cycles until fracture)

$N_{min} \leq 5000$

No torsion testing

Values of $k > 15$ are excluded

Steam treated and carbonitrided specimens are excluded

Hardness is given in Vickers scale

After filter criteria were established 410 S–N curves remain with a total number of 12,966 specimens tested in finite life regime. The number of specimens used for determining the behavior in finite life regime in each S–N curve varies between 10 and 82, whereas the mean value is 27 specimens. The mean value for the slope can be calculated as $k_{m} = 7.19$ and $N_{k, m} = 1.16 \times 10^{6}$ .

Results and discussion

The functional relationships described above can be used serially to estimate the locally tolerable stress amplitude of arbitrarily shaped and loaded components for a probability of failure of 50%. First, the reference fatigue strength for an unnotched specimen with $V_{90} \approx 10$ mm $^{3}$ is determined based on hardness and density using equations (4), (5), (8), and (7). Equation (10) allows the conversion of the reference fatigue strength to different part geometries in combination with any load case with $V_{90} \neq 10$ mm $^{3}$ . By determining the mean stress sensitivity, the Haigh diagram can then be constructed, enabling the determination of fatigue strength for various stress ratios. To verify the calculation procedure, fatigue strengths were calculated for all data sets in the database where the hardness is available in the Vickers scale or can be converted to it. The fatigue strengths were calculated based on hardness, density, highly loaded volume, and stress ratio. Fatigue strengths obtained from specimens with surface treatments such as shot peening were not considered. Since the hardness at a density of 7.0 g/cm $^{3}$ was not available for the data sets, it was estimated using equation (5) to subsequently determine the characteristic parameter $S_{A 7.0} / H_{7.0}$ . For newly developed or unknown materials, $H_{7.0}$ must be measured.

Figure 5 presents the calculated values compared to the experimental values from the database. The data scatters a lot but most values lie within the $\pm 15 %$ range. Figure 6 shows the histogram of deviations in percentage. As the mean value is nearly zero, it suggests that there is no systematic underestimation or overestimation of $σ_{A}$ .

Figure 5.

Comparison of calculated values with experimental data.

Figure 6.

Histogram of the percentage deviations between the calculated and experimentally determined fatigue strengths.

In our own investigations in Burkamp et al.¹⁸¹ it was shown in a larger statistical evaluation that for failure probability deviating from $P = 50 %$ a logarithmic standard deviation of $s_{\log} = 0.025$ in the direction of stress amplitude or respectively a spreading margin $T_{S} = σ_{A 90} / σ_{A 10} = 1.16$ can be assumed. With this $σ_{A}$ can be determined for arbitrary probabilities of failure. Using the above stated values of $k = 7$ and $N_{k} = 1 \times 10^{6}$ to draw the finite life in the S–N diagram, a complete synthetic S–N curve can be constructed for any component. Figure 7 shows the parameters in a sketched S–N curve.

Figure 7.

Schematic representation of the synthetic S–N curve with characteristic values.

Conclusions

In this study, with inclusion of results from Burkamp et al.¹⁸¹ and Beiss et al.,¹⁶⁷ a novel algorithm is presented, which enables the construction of synthetic S–N curves for components made of sintered steels with arbitrary shapes and loading conditions, given the knowledge of density and hardness at the proof point. The algorithm allows the density and hardness to be measured for unknown alloys and an alternating strength to be estimated from these, which can then be transferred to a component under a specific load. Alternatively, the fatigue limit under fully reversed loading can be determined experimentally on an unnotched specimen to start the calculation with a experimentally validated fatigue limit for a fully reversed load case.

Footnotes

Acknowledgments

Most of this work was financially supported by the Federal Ministry of Economics and Energy of Germany under AIF-No. 20234 N/1 via Forschungskuratorium Maschinenbau (FKM). The final report for the project can be found in Burkamp et al.¹⁸⁰ All this support is gratefully acknowledged.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iD

Tobias Hajeck

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