Appendix (references) on Fatigue Analysis

The stress cycle. The general properties

After a certain number of repeated loading (or stress cycles) the final destruction of parts can occur,  on the other hand, it is possible that the destruction does not take place under constant stress load.The number of stress cycles until the fracture depends on the magnitude (amplitude of the stress) and varies over a wide range. For destroying under high stresses 5-10 cycles are enough, on the other hand, under lower stresses detail withstand millions or billions of cycles, and under even less stresses it is able to operate indefinitely.

There are maximum σmax and minimum σmin stress cycles by which to understand the largest and smallest of the algebraic value of the stress cycle. As a medium-stress σm and amplitude σa of the stress cycle take: σm= ( σmax + σmin ) / 2 , σa= ( σmax - σmin ) / 2. Scale of stress is called the difference between the maximum and minimum stress cycle, i.e. 2σa= ( σmax - σmin ).

The cycle in which the maximum and minimum stress in absolute value are different, called asymmetrical.

A special case of the asymmetric cycle is pulsating cycle at which the minimum stress cycle is zero: σmin = 0.

Symmetric is a cycle in which the maximum and minimum stresses are equal in magnitude but opposite in sign.

To characterize the degree of asymmetry of stress is used the stress asymmetrical ratio, which is the ratio of minimum to maximum stress cycle: R=σmin/σmax .

Methods  of stress correction

Let σ* is corrected alternate stress, σY  is yield stress, σT is tensile strength, then:

1.Soderberg method
σ* = σa / ( 1 - σm / σY )

2.Goodman method is used for brittle materials:
σ* = σa / ( 1 - σm / σT )

3.Gerber method is used for plastic materials:
σ* = σa / ( 1 - ( σm / σT )2 )

Evaluation of characteristics of fatigue resistance under complex stress state

Strength conditions with alternating stresses in general terms similar to the strength conditions of static analysis, but as the maximum permissible stress used fatigue limit σR . Accordingly, hypothesis testing fatigue strength criterion to avoid plastic deformation in the fatigue calculations take the form σ* ≤ σR . We recall the general expressions of traditional hypothesis test of strength (for the plasticity condition), used to assess the strength of structures:

1.Tresca - Saint-Venant hypothesis (the hypothesis of maximum shear stress)
σY < ( σ1 - σ3 ), | σ1 | ≥ | σ2 | ≥ | σ3 |

2.Huber - Mises hypothesis  (distortion energy hypothesis)

3.Mohr hypothesis (hypothesis the greatest principal stress)
σY < σ1 , | σ1 | ≥ | σ2 | ≥ | σ3 |

where σY  is yield stress, σ1 , σ2 , σ3 are the principal stresses. Therefore, when analysing the fatigue, we also get three options for the safety factor corresponding to each of the generally accepted theory of material strength.

 

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