Basquin equation

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The Basquin equation of materials engineering (after Olin Hanson Basquin , 1910) describes the course of the Wöhler line in a double logarithmic representation in the area of ​​the fatigue strength as a straight line, i.e. between 10 ⁴ and 10 ⁶ cycles . The representation takes place via a power law that links the load amplitude with the number of cycles.

Preview

When performing fatigue strength tests in which specimens or components are subjected to a periodically changing load, they can fail prematurely or they go through the test completely. The latter is also referred to as passers-through.

If the fatigue strength test was carried out according to the pearl string method, test results are available on several load horizons. The resulting Wöhler line can be approximated as a straight line (long-term strength line) in the double-logarithmic representation in the fatigue strength range. The position and slope of the fatigue strength straight line are described by the Basquin equation: ${\displaystyle C}$ ${\displaystyle k}$

${\displaystyle N=C\cdot {L_{a}^{-k}}}$

With

${\displaystyle N}$ - number of cycles,

${\displaystyle C}$ - constant to describe the fatigue strength straight line,

${\displaystyle L_{a}}$ - amplitude of a load variable (force, tension, displacement),

${\displaystyle k}$ - Inclination of the fatigue strength straight line.

By taking the logarithm and converting the Basquin equation into a straight line equation

${\displaystyle logN=logC-k\cdot {logL_{a}}}$

The parameters and can be determined using regression analysis . ${\displaystyle C}$ ${\displaystyle k}$

Stress Wöhler curve

The number of cycles up to failure is plotted in a Wöhler diagram as a function of the stress amplitude. Basquin recognized that the Wöhler curve with pure alternating stress ( ) takes a linear course from a single load to fatigue strength if the true stress amplitudes and number of cycles are plotted logarithmically. ${\displaystyle N}$ ${\displaystyle \sigma _{A}}$ ${\displaystyle \sigma _{m}=0}$

With the transformed Basquin equation, the following relationship applies to pure alternating loads

${\displaystyle \sigma _{A}=\sigma _{f}^{'}(2N)^{b}}$.

With

${\displaystyle \sigma _{A}}$ - true stress amplitude in [MPa],

${\displaystyle \sigma _{f}^{'}}$ - fatigue strength coefficient in [MPa],

${\displaystyle 2N}$- number of reversals of the load until breakage ,

${\displaystyle b}$ - fatigue strength exponent in [-]

This equation is based on the number of load reversals (1 cycle equals 2 reversals), the fatigue strength coefficient, and the fatigue strength exponent , the latter of which are each based on an inversion, not a cycle. The fatigue strength coefficient corresponds almost to the true breaking stress in the tensile test. A rough guide value applies to unalloyed and low-alloy steels and to aluminum and titanium alloys . The fatigue strength exponent depends on many factors. For most materials, unnotched specimens have a value between −0.05 and −0.12. ${\displaystyle 2N}$ ${\displaystyle \sigma _{f}^{'}}$ ${\displaystyle b}$ ${\displaystyle \sigma _{f}^{'}}$ ${\displaystyle \sigma _{f}^{'}=R_{m}\cdot 1{,}5}$ ${\displaystyle \sigma _{f}^{'}=R_{m}\cdot 1{,}67}$ ${\displaystyle b}$

A double logarithmic plot results in a falling straight line, as shown in Figure 2. The true stress amplitudes are plotted on the ordinate axis and the number of cycles on the abscissa axis in a logarithmic scale. The fatigue strength occurs with cycles. This corresponds to a load reversal of . ${\displaystyle \sigma _{A(W)}}$ ${\displaystyle N}$ ${\displaystyle N>10^{6}}$ ${\displaystyle log\ 2N>6{,}3}$

However, the equation is purely empirical and has no “real” physical background, since the plastic strain amplitudes actually cause damage to the microstructure of the material and thus a reduction in service life, see Coffin-Manson model .

For long lifetimes, however, the plastic amplitudes are so small and difficult to detect using measurement technology that the lifetime is often determined in a voltage-controlled manner, especially in the HCF ( high-cycle fatigue ) range. The Basquin equation has proven to be advantageous here.

Extension for the strain Wöhler curve

By using Hooke's law, the following relationship applies

${\displaystyle \sigma _{A}=E\cdot \varepsilon _{A,el}}$.

With Hooke's law and the Basquin equation for the stress-Wöhler curve, the relationship between the number of load reversals and the elastic strain amplitude can be obtained by rearranging and combining

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

With

${\displaystyle \varepsilon _{A,el}}$ - elastic strain amplitude in [-],

${\displaystyle \sigma _{f}^{'}}$ - fatigue strength coefficient in [MPa],

${\displaystyle {E}}$ - modulus of elasticity in [MPa],

${\displaystyle 2N}$- number of reversals of the load until breakage ,

${\displaystyle b}$ - fatigue strength exponent in [-]

This printout can be used to create a strain Wöhler curve (see basic notch concept ).

Individual evidence

1. ^ OH Basquin: The exponential law of endurance tests. In: Proc. ASTM . 11, 1910, p. 625.
2. ↑ DIN 50100: Fatigue strength test - implementation and evaluation of cyclical tests with constant load amplitude for metallic material samples and components , DIN German Institute for Standardization eV, 2016.
3. ↑ Ralf Bürgel, Hans Jürgen Maier, T. Niendorf: Handbook high temperature materials technology, fundamentals, material stresses, high temperature alloys and coatings. Vieweg + Teubner Verlag, 2011, ISBN 978-3-8348-1388-6 .
4. ^ S. Lampman: ASM Handbook. Volume 19: Fatigue and Fracture. ASM International, 1996, ISBN 0-87170-385-8 .
5. ↑ Ralf Bürgel, HJ Maier, T. Niendorf: Handbook high temperature materials technology, basics, material stresses, high temperature alloys and coatings. Vieweg + Teubner Verlag, 2011, ISBN 978-3-8348-1388-6 .
6. ↑ Dieter Radaj, M. Vorwald: Fatigue Strength Basics for Engineers. Springer-Verlag, 2007, ISBN 978-3-540-44063-5 .
7. ↑ Erwin Haibach: Fatigue strength method and data for component calculation. Springer-Verlag, 2006, ISBN 3-540-29363-9 .