Notch basic concept

The notch basic concept , also notch elongation concept , notch elongation verification is a concept in materials testing according to which technical components are tested and evaluated for their fatigue strength ( material fatigue ). It is a local, local concept for calculations and tests that is only meaningful for certain points on the component. The bottom of the notch is considered , the most highly stressed point at which the component can break under tension or strain.

The notch basic concept is used in addition to the material properties when the fatigue strength of welded and non-welded components is to be assessed, which are cyclically, i.e. often repeatedly loaded, in the same way as part of an operational strength verification . The component test is not passed if a " technical crack " with a crack length of more than approx. 0.5 mm occurs on the surface before the existing number of load cycles, this is a failure criterion for the strength verification . A transferability of the uniaxial, cyclically elastic - plastic material - and damage behavior to the most highly stressed part of the component, i.e. H. in the bottom of the notch.

Background and functionality

When assessing the fatigue strength and the service life of technical components , the associated stress-strain hystereses are determined on the basis of elastic-plastic notch stresses and strains in the notch base and, to put it simply, compared with the material-based strain Wöhler curve .

The notch base concept is based on the idea that the mechanical behavior in the notch base is comparable to that of a miniaturized, uniaxially loaded, unnotched (or slightly notched) test specimen in relation to the global deformation and damage behavior (material-based concept). The influence of the supporting effect is taken into account on the basis of an increased Wöhler expansion curve or a notch elongations with a reduced number of supports in conjunction with an unchanged Wöhler expansion curve.

The concept includes the notch-mechanical expansion of the material samples based notch stress detection of the fatigue strength to the areas of time and durability , taking the place of the elastic notch stresses occur, the elastic-plastic elongations notch as the main stress parameter. In the short-term strength range, the plastic elongation component is primarily determining the fatigue strength. A special feature of the notch strain concept is that, in contrast to other methods ( nominal stress concept , structural stress concept , notch stress concept ), for which component Wöhler lines must always be known, these ( component Wöhler lines ) can be calculated during the notch strain analysis.

Knowledge of the following input variables and the implementation of the following calculation steps are required to carry out a strength verification using the notch strain concept:

• analytical description of the strain Wöhler curve according to Manson, Coffin, Morrow
• analytical description of the material law with cyclic loading, with the cyclically stabilized material flow curve according to Ramberg and Osgood
• Consideration of material behavior under cyclic loading, masing behavior and memory laws
• Calculation of elastic-plastic stresses and strains, e.g. B. by means of a notch approximation relationship according to Neuber
• Consideration of medium stresses at operating load by means of a damage parameter, e.g. B. P _SWT according to Smith, Watson and Topper
• Consideration of the macro-support effect due to elastic-plastic stresses and strains, e.g. B. Extension of the stress gradient approach, according to Siebel and Stieler, to include Neuber's macro-supportive effect due to plastic strain components in the notch base
• Consideration of other factors such as surface factor or residual stresses, v. a. production-related influencing factors
• Damage calculation , e.g. B. linear damage accumulation .

Application of the notch elongation concept

Strain Wöhler curve

Figure 1: Strain Wöhler curve (schematic) with associated parameters

The basis for describing the resistance of the material is the strain Wöhler curve (Figure 1: strain Wöhler curve ).

This is determined with a polished sample in a strain- controlled test , with a constant mean stress ratio, normally , i. H. medium voltage free. The relationship between the strain amplitude and the number of cycles can be expressed according to Manson, Coffin and Morrow, for a certain mean stress ratio, in the following form: ${\displaystyle R_{\varepsilon }=-1}$

${\displaystyle \varepsilon _{A}=\varepsilon _{A,el}+\varepsilon _{A,pl}={\frac {\sigma _{f}^{'}}{E}}\cdot (2N)^{b}+\varepsilon _{f}^{'}\cdot (2N)^{c}}$

With

${\displaystyle \varepsilon _{A}}$ - strain amplitude,

${\displaystyle N}$- oscillation cycles ( oscillation cycles = inversion of oscillation), ${\displaystyle N}$ ${\displaystyle 2N}$

${\displaystyle \sigma _{f}^{'}}$ - fatigue strength coefficient,

${\displaystyle b}$ - fatigue strength exponent,

${\displaystyle \varepsilon _{f}^{'}}$ - cyclic ductility coefficient,

${\displaystyle c}$ - cyclic ductility exponent,

${\displaystyle E}$ - Modulus of elasticity.

Material flow curve with cyclic loading

Figure 2: Stress-strain hystereses with the associated load-time sequence, "Calculation example notch basic concept"

A (cyclically stabilized) material law is required to determine the over-elastic stresses or strains in the notch base . From an experimental point of view, the cyclical material flow curve is obtained from the strain-controlled Wöhler curve experiment. In order to determine the cyclic stress-strain curve faster and less time-consuming than point-by-point by means of strain-controlled Wöhler tests, there is the incremental step test, in which the strain amplitude is gradually increased up to a specified maximum value and then gradually reduced again. Any hardening or softening of the material takes place in a similar, but not necessarily the same way as in the Wöhler test. After two or three up-and-down sequences of this type, the reversal points of the stress provide a largely stabilized cyclic stress-strain curve that roughly corresponds to that obtained from Wöhler tests. For the analytical description of the material flow curve, z. B. the approach of Ramberg and Osgood can be used. The result is, for any load-time sequence, stress-strain hystereses with corresponding open and closed hysteresis branches in a - diagram (Figure 2: stress-strain hysteresis ). The cyclically stabilized - curve is written (for the first load) as follows: ${\displaystyle \sigma }$ ${\displaystyle \varepsilon }$ ${\displaystyle \sigma }$ ${\displaystyle \varepsilon }$

Figure 3: "Universal Material Law" (UML) according to Bäumel and Seeger

${\displaystyle \varepsilon =\varepsilon _{el}+\varepsilon _{pl}={\frac {\sigma }{E}}+\left({\frac {\sigma }{K^{'}}}\right)^{1/n'}}$,

where represent the cyclic hardening coefficient and the cyclic hardening exponent. The following relationships also apply between the material flow curve and the strain Wöhler curve (compatibility relationship): ${\displaystyle K^{'}}$ ${\displaystyle n'}$

${\displaystyle K^{'}={\frac {\sigma _{f}^{'}}{(\varepsilon _{f})^{n'}}}}$

${\displaystyle n'={\frac {b}{c}}}$

The "Universal Material Law" (UML) by Bäumel and Seeger offers a possibility of estimating cyclical material parameters for unalloyed and low-alloy steels as well as for aluminum and titanium alloys (Fig. 3: Universal Material Law )

Masing behavior and material memory

In the case of cyclical loading of technical components (i.e. multiple reloading), two further effects must be taken into account: the so-called masing behavior and the memory laws (or memory effects). Masing behavior means that, when reloading a voltage or strain range , is set which corresponds to twice the amplitude value of the voltage or elongation of the Erstbelastungskurve (Figure 4: Three different forms of "Werkstoffgedächtnisse" ): ${\displaystyle (\Delta \sigma }$ ${\displaystyle \Delta \varepsilon )}$

${\displaystyle \Delta \sigma =2\cdot \sigma }$, . ${\displaystyle \Delta \varepsilon =2\cdot \varepsilon }$

Taking into account the masing behavior, the following results for the material flow curve according to Ramberg and Osgood with reloading:

${\displaystyle \Delta \varepsilon ={\frac {\Delta \sigma }{E}}+2\cdot \left({\frac {\Delta \sigma }{2\cdot K^{'}}}\right)^{1/n'}}$.

The memory laws (memory effects) are a kind of material memory. According to, three different forms of memory laws can be distinguished (Memory 1 to Memory 3, see picture below). The following description of the memory effects was taken from:

• For the first load, the stress-strain curve is used as the stress-strain path (cyclically stabilized - curve or first load curve, path 0–1). ${\displaystyle \sigma }$ ${\displaystyle \varepsilon }$
• Memory 1: After closing a hysteresis loop (reloading curve, taking into account the masing behavior, - curve) that was started on the first loading curve (path 1-2-1), the stress-strain path continues on the first loading curve (path 1-3). ${\displaystyle \Delta \sigma }$ ${\displaystyle \Delta \varepsilon }$
• Memory 2: After closing a hysteresis loop that was started on a loop branch (path 4-5-4), the stress-strain path follows the original loop branch (path 3-4-6).
• Memory 3: A hysteresis loop branch (path 3-4-6) started on the first load curve ends when the mirror point 6 of its starting point 3 in the opposite quadrant is reached; then the stress-strain path continues on the initial load curve (path 6-7).

Image 4: Three different forms of "material memories"

Notch approximation according to Neuber

Figure 5: Determination of the local stresses and strains using the Neuber rule

The relationship between the external load and the local stress or strain can be established as follows using the notch approximation relationship according to Neuber (Fig. 5: Determination of the local stresses ), based on the macro-support formula.

${\displaystyle K_{t}^{2}=K_{\varepsilon }\cdot K_{\sigma }}$ (Macro support formula),

thereby make the (theoretical elasticity) notch shape figure , and the inelastic stress or strain concentration factors. Under consideration of the Hooke's Law , as well as the relationship between the local (inelastic) strain or stress and mechanical stress and elongation ( ) leaves the Neuber - Write the formula as follows: ${\displaystyle K_{t}}$ ${\displaystyle K_{\varepsilon }}$ ${\displaystyle K_{\sigma }}$ ${\displaystyle (E={\frac {\sigma }{\varepsilon }})}$ ${\displaystyle K_{\varepsilon }={\frac {\varepsilon }{\varepsilon _{n}}},K_{\sigma }={\frac {\sigma }{\sigma _{n}}}}$

${\displaystyle {\frac {K_{t}^{2}\cdot \sigma _{n}^{2}}{E}}=\varepsilon \cdot \sigma }$.

Target due to the supporting effect of the stress be reduced, instead of the notch form factor , the notch effect number used, and the above-described material law are equated: ${\displaystyle n}$ ${\displaystyle K_{t}}$ ${\displaystyle K_{f}}$

${\displaystyle {\frac {K_{f}^{2}\cdot \sigma _{n}^{2}}{E}}=\left({\frac {\sigma }{E}}+\left({\frac {\sigma }{K^{'}}}\right)^{1/n'}\right)\cdot \sigma }$ (First load)

${\displaystyle {\frac {K_{f}^{2}\cdot \Delta \sigma _{n}^{2}}{E}}=\left({\frac {\Delta \sigma }{E}}+2\cdot \left({\frac {\Delta \sigma }{2\cdot K^{'}}}\right)^{1/n'}\right)\cdot \Delta \sigma }$ (Reloading, masing behavior)

Strictly speaking, the Neuber rule only applies when the net cross-section is purely elastic. The elastic-plastic notch stress after partial or complete plasticization of the net cross-section can be determined according to Seeger and Heuler using the following modification of the Neuber formula:

${\displaystyle K_{t}^{2}\cdot {\frac {\varepsilon _{n}^{*}\cdot E}{\sigma _{n}^{*}}}=K_{\varepsilon }\cdot K_{\sigma }}$.

The load-related nominal voltage

${\displaystyle \sigma _{n}^{*}=\sigma _{n}\cdot {\frac {K_{t}}{K_{p}}}}$,

together with the nominal elongation modified according to the stress-strain curve of the material

${\displaystyle \varepsilon _{n}^{*}=f(\sigma _{n}^{*})}$

and the load capacity number

${\displaystyle K_{p}={\frac {\sigma _{pl}}{\sigma _{n,F}}}}$

introduced. The modified nominal expansion is written as follows when using the material law according to Ramberg and Osgood (see above):

${\displaystyle \varepsilon _{n}^{*}={\frac {\sigma _{n}^{*}}{E}}+\left({\frac {\sigma _{n}^{*}}{K^{'}}}\right)^{1/n'}}$

Medium stress influence and damage parameters

With the calculation modules discussed so far (masing behavior, material memory and the notch approximation relationship), the stress-strain paths can be determined taking into account the material behavior and the load acting on the component. If there is a single-stage load, the damage-effective stress-strain hystereses can be compared directly with the strain Wöhler curve. The area within a closed hysteresis can be viewed as a measure of the material damage. In the case of an operating load in which any load-time sequence acts on the component in terms of time, a comparison of the hystereses with the strain Wöhler curve, which applies to a certain load ratio, is no longer possible. Here the hystereses must be transformed with respect to the mean stress or strain, i.e. This means that they can be converted into damage- equivalent hystereses with a mean voltage of . This is done with a so-called damage parameter (or also called medium-voltage parameter), which takes this fact into account by taking it into account in the damage accumulation. The most widely used parameter in the literature comes from Smith, Watson and Topper: ${\displaystyle \sigma _{m}=0}$

${\displaystyle P_{SWT}={\sqrt {(\sigma _{a}+\sigma _{m})\cdot \varepsilon _{a}\cdot E}}}$.

Here , or correspond to the amplitude or the mean value of a stress-strain hysteresis. A weakness of the damage parameter P _SWT is that it predicts the tolerable amplitudes, especially with high-strength steels, significantly too high for pressure medium stresses ( ). An overview of the various damage parameters is mainly given in. ${\displaystyle \sigma _{a}}$ ${\displaystyle \varepsilon _{a}}$ ${\displaystyle \sigma _{m}}$ ${\displaystyle \sigma _{m}<0}$

On the side of the load-bearing capacity, the corresponding strain Wöhler curve must be converted into a Wöhler curve corresponding to the damage parameter. Using the approach according to Manson, Coffin, Morrow (see above), the P _SWT -Wöhler line is obtained for the medium voltage : ${\displaystyle \sigma _{m}=0}$

${\displaystyle P_{SWT}={\sqrt {\sigma _{f}^{'2}\cdot (2\cdot N)^{2b}+\sigma _{f}^{'}\cdot \varepsilon _{f}^{'}\cdot E\cdot (2\cdot N)^{b+c}}}}$.

Notch sensitivity and macro support

In the case of oscillating loads, the notch effect is taken into account by means of the notch effect number in the strength analysis (the formula symbol is often used). The possible range of values is: ${\displaystyle K_{f}}$ ${\displaystyle \beta _{k}}$ ${\displaystyle K_{f}}$

${\displaystyle 1\leq K_{f}\leq K_{t}}$

If the amount of the notch action number reaches the value of the (theoretical elasticity) notch shape number (or ) the theoretically calculated stress peak is fully damaging; if, on the other hand, the notch action number is 1, this means that the notch does not reduce the strength of an unnotched test bar with the dimensions of the notch cross-section. The ratio between the theoretical and effective stress increase is expressed by means of a supporting number : ${\displaystyle K_{t}}$ ${\displaystyle \alpha _{k}}$

${\displaystyle n={\frac {K_{t}}{K_{f}}}}$.

Figure 6: Stress gradient approach according to Siebel and Stiehler

There are various models for determining the support number ; two well-known models are:

• the Siebel and Stieler method (stress gradient approach)
• as well as the voltage averaging approach according to Neuber

For further information, see Basics or further specialist literature.

The calculation of the supporting number according to Siebel and Stieler - this is to be briefly addressed here - is based on the related stress gradient at the point of greatest stress. The stress gradient related to the maximum elastic notch stress is calculated as follows (Figure 6: stress gradient approach ): ${\displaystyle \chi ^{*}}$

${\displaystyle \chi ^{*}={\frac {1}{\sigma _{max}}}\left({\frac {d\sigma }{dx}}\right)_{x=0}}$.

The assignment between the related stress gradient and the dynamic support number (the letter in the subscript indicates that the dynamic support number, see above, was determined according to the stress gradient approach) depends on the material and can be taken from tables or specialist literature (e.g.) become: ${\displaystyle \chi ^{*}}$ ${\displaystyle n_{\chi ^{*}}}$ ${\displaystyle \chi ^{*}}$

${\displaystyle n_{\chi ^{*}}=f(\chi ^{*}}$, Material . ${\displaystyle )}$

The support figure determined in this way cannot be used directly within the framework of the notch expansion concept. This must first be modified due to possible proportions of larger plastic deformations, which are based on the macroscopic supporting effect (reduction of the notch effect through local flow). An approach that comes from Neuber and is described in, results in the macro support effect : ${\displaystyle n_{p}}$

${\displaystyle n_{p}={\sqrt {1+{\frac {\varepsilon _{D}^{pl}}{\varepsilon _{D}^{el}}}}}}$,

where and are the elastic and plastic strain components in the fatigue strength, which are taken from the strain Wöhler curve. The modified support number with regard to the macro support effect results in: ${\displaystyle \varepsilon _{D}^{el}}$ ${\displaystyle \varepsilon _{D}^{pl}}$ ${\displaystyle n}$

${\displaystyle n={\frac {n_{p}}{n_{\chi ^{*}}}}}$.

The supporting effect can now either be on the side of the load, in which the notch effect number is calculated using the supporting number described here and used to determine the local stresses and strains (see section Notch approximation according to Neuber ), or alternatively, on the side of the loadability , by the Raising the P _SWT -Wöhler curve, the following must be taken into account:

${\displaystyle P_{SWT,n}=n\cdot P_{SWT}}$

Surface factor

Basically, all significant factors and influences must be taken into account in the service life calculation. In practice, these are mainly production-related influences such as B. residual stresses from surface hardening , surface roughness or a steep stress gradient with sharp notches .

To z. For example, to include the influence of surface roughness in the calculation, there are several options. On the one hand, the strain Wöhler curve can be determined on samples with a correspondingly rough surface. On the other hand, the damage parameter Wöhler curve can be reduced using a surface factor in the ratio of the fatigue strengths for rough and polished surfaces. In both cases i. d. As a rule, in addition to the roughness influence, the influence of any residual stresses, which can be assessed in another way, is included . See z. B. the FKM guideline .

Damage calculation

Figure 7: Linear damage accumulation

The damage calculation can be carried out on the basis of the calculated stress-strain hysteresis . The damage contribution of each individual closed stress-strain hysteresis is accumulated to a damage total: ${\displaystyle D_{j}}$ ${\displaystyle D}$

${\displaystyle D=\sum _{j=1}^{N_{i}}D_{j}=\sum _{j=1}^{N_{i}}{\frac {1}{N_{j}}}=1.0}$,

with ( miner's rule ) or another empirically proven damage sum ( relative miner's rule ). See also Figure 7 and the article: Linear damage accumulation . ${\displaystyle D=1.0}$

See also

High frequency impact treatment ; Weld seam aftertreatment process to extend service life by rounding, smoothing and strengthening the surface layer of the notch at the seam transition and introducing residual compressive stresses.

Individual evidence

1. ↑ ^{a b c} cf. Dieter Radaj, Cetin M. Sonsino, Wolfgang Fricke: Fatigue assessment of welded joints by local approaches . 2nd Edition. Woodhead, Cambridge 2006, ISBN 1-85573-948-8 .
2. ↑ ^{a b} cf. S. Greuling, T. Seeger: Concepts for determining the fatigue strength of thin sheet metal welds . In: MP Materials Testing . tape 04/2007 , 2007, p. 157-169 .
3. ↑ ^{a b c d e} Dieter Radaj, Michael Vormwald: Fatigue strength: Fundamentals for engineers . 3. Edition. Springer, Berlin / Heidelberg 2007, ISBN 978-3-540-71459-0 .
4. ↑ ^{a b c d e f g h} Erwin Haibach: Durability . 2006, ISBN 3-540-29363-9 .
5. ^ SS Manson: Fatigue: A complex subject - Some simple approximations . In: Experimental Mechanics . tape 5 , no. 4 , July 1965, p. 193-226 , doi : 10.1007 / BF02321056 .
6. ↑ LF Coffin, jr .: A Study of the Effects of Cyclic Thermal Stresses on a Ductile Metal . In: Trans. ASME . tape 76 , 1954, pp. 931-950 .
7. ↑ Jodean Morrow: Cyclic Plastic Strain Energy and Fatigue of Metals . In: Bj Lazan (Ed.): Internal Friction, Damping, and Cyclic Plasticity . ASTM International, ISBN 978-0-8031-6160-3 , pp. 45-87 , doi : 10.1520 / STP43764S .
8. ^ R. Hales, SR Holdsworth, MP O'Donnell, IJ Perrin, RP Skelton: A Code of Practice for the determination of cyclic stress-strain data . ( Memento of the original from November 4, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF; 255 kB) June 21, 2012 
9. ^ W. Ramberg, WR Osgood: Description of stress-strain curves by three parameters . In: NACA Technical note . tape 902 , no. 902, 1943, pp. 1-28 .
10. ↑ A. Bäumel, Jr and T. Seeger: Materials data for cyclic loading, supplement 1 . In: Elsevier . 1990, ISBN 978-0-444-88603-3 .
11. ↑ G. Masing: Internal stresses and solidification in brass . In: Proc. 2nd Int. Cong. of Appl. Mech. Zurich 1926, p. 332-335 .
12. ↑ M. Matsuishi, T. Endo: Fatigue of metals subjected to varying stress . In: Proc. Kyushu Branch of Japan Society of Mechanical Engineers, Fukuoka, Japan . 1968, p. 37-40 .
13. ↑ UH Clormann, T. Seeger: Rainflow-HCM - A counting method for operational stability based on material mechanics . In: Steel construction . tape 55 , no. 3 , 1986.
14. ^ H. Neuber: Theory of Stress Concentration for Shear-Strained Prismatical Bodies With Arbitrary Nonlinear Stress-Strain Law . In: Journal of Applied Mechanics . tape 28 , no. 4 , 1961, pp. 544 , doi : 10.1115 / 1.3641780 .
15. ↑ T. Seeger, P. Heuler: Generalized application of Neuber's rule . In: Journal of Applied Mechanics . No. 8 , 1980, p. 199-204 .
16. ↑ KN Smith, P. Watson, TH Topper: A stress-strain function for the fatigue of metals . In: Journal of Materials . tape 5 , no. 4 , 1970, pp. 767-778 .
17. ↑ ^{a b} S. Greuling: Fatigue strength calculation of auto-fretted internal pressure loaded components with bore intersections, taking into account permanent cracks . In: Publications of the Institute for Steel Construction and Mechanics of Materials at the Technical University of Darmstadt . 2005, ISBN 978-3-939195-01-6 .
18. ↑ D. Radaj, CM Sonsino: Fatigue strength of welded joints according to local concepts . DVS Media, specialist book series Schweisstechnik Volume 142, Düsseldorf 2000, ISBN 978-3-87155-191-8 .
19. ↑ ^{a b c} L. Issler, H. Ruoss, P. Häfele: Strength theory - basics . Springer textbook, 2005, ISBN 978-3-540-40705-8 .
20. ↑ E. Siebel, M. Stieler: Uneven stress distribution with oscillating stress . No. 5 . VDI-Z. 97, 1955, pp. 121-126 .
21. ↑ H. Neuber: About the consideration of stress concentration in strength calculations . No. 7 . Construction 20, 1968, p. 245-251 .
22. ↑ ^{a b} Research Board of Mechanical Engineering :: Computational proof of strength for machine components . 5th expanded edition. VDMA-Verlag, 2003, ISBN 3-8163-0479-6 .