# Investigation of Changes in Fatigue Damage Caused by Mean Load under Block Loading Conditions

^{*}

## Abstract

**:**

_{m}= 0 (symmetric loads), the ratio of the total degree of fatigue damage varies from 10% for ε

_{a}= 0.2% to 3.5% for ε

_{a}= 0.6%. The largest differences in the calculation for ratios of the partial degrees of fatigue damage were observed in relation to the reference case for the sequence of block n

_{3}, where ε

_{m}= 0.4%. The simulation results show that higher mean strains change the properties of the material, and in such cases, it is necessary to take into account the influence of the mean value on the material response under block loads.

## 1. Introduction

_{i}/N

_{f}represents the proportion of cycles n

_{i}with respect to the life of the N

_{f}sample. The index i (here, i = 1, 2) denotes the step in the load block.

_{1}, the material is loaded with stress σ

_{1}and, then, the second phase of the number of cycles n

_{2}with stress σ

_{2}is applied. Such tests are carried out in two ways:

- After the first phase of loading, the second phase of testing is carried out to obtain the criterion of sample failure.
- The sequence of cycles n
_{1}and n_{2}is determined, and this load sequence is repeated until the criterion of sample failure is not reached.

_{i}is the number of cycles in a load block with stress σ

_{i}, and N

_{i}is the number of cycles to failure determined from the base fatigue characteristics under constant-amplitude loads.

## 2. Stress–Strain Analysis Model

_{m}and mean stress σ

_{m}(see Figure 3).

_{m}

_{2}represents the deformation related to the state of the material previously achieved. However, to take into account the previous load history, it was assumed that the starting point for the deformation analysis is the level of permanent deformation after the previous load blocks. Hence, for further analysis, the strain was marked as ε

_{m}

_{2z}.

## 3. Analytical Simulation

_{m}

_{1}= 0, ε

_{m}

_{2}= 0.2%, and ε

_{m}

_{3}= 0.4%. Each load block consisted of n

_{1}= n

_{2}= n

_{3}= 1000 cycles. The strain amplitude was made to vary in the range ε

_{a}= 0.2–0.60%. The fatigue life N was estimated by using the approximate relationship log(N) = −12.88·log(σ

_{a}) − 36.38 on the basis of the data presented by Lei et al. [20].

_{i}= n

_{i}/N

_{i}was computed (i = 1, 2, and 3 for each block load sequence), where N

_{i}is the expected fatigue life for the obtained stress state, assuming a constant-amplitude load. Then, the total degree of the damage accumulation was calculated as a sum of the partial degrees.

- -
**Case A**, where the change in K′ and n′ parameters was taken into account for different values of the mean strain e_{m},- -
**Case B**, where the influence of mean strain e_{m}on the values of the coefficients K ‘and n′ was omitted, assuming their values to be e_{m}= 0.

**Case A**and

**Case B**are listed. In the tables, indices 1, 2, and 3 refer to n

_{1}, n

_{2}, and n

_{3}in Figure 6, while letters A and B refer to Cases A and B.

_{iB}/D

_{iA}of the partial degrees of fatigue damage and the ratio D

_{B}/D

_{A}of the total degree of fatigue damage for both cases are shown.

_{m}= 0.4%. Meanwhile, the ratio D

_{B}/D

_{A}of the total degree of fatigue damage varied from −10% to 3.5%. The biggest inaccuracies in fatigue life estimations were expected for stress states approaching the fatigue limit. Thus, under these conditions, the influence of mean strain value cannot be neglected.

_{1}, n

_{2}, and n

_{3}(see Figure 6) depending on the mean strain. By increasing the value of the K′ coefficient, it can be noted that the largest changes in relation to the reference case (Case A, Table 3) were observed for the third load sequence n

_{3}, where the mean strain value was the largest (ε

_{m}= 0.4%). For K′ = 722.6 MPa, the ratio of the total degree of fatigue damage was D/D

_{A}= 1.5–17, which gives a good approximation to the reference case. In the case of K′ = 587.7 MPa, the accumulated fatigue damage was four times smaller than the reference value.

_{1}, n

_{2}, and n

_{3}.

_{A}= 0.9–1.04) and the partial degrees of fatigue damage. Decreasing the value of n′ for cases S5 and S6 caused a decrease in the effect on the accumulation of fatigue damage for the n

_{3}block sequence (ε

_{m}= 0.4%). For the S5 sequence, the total damage ratio was determined to be D/D

_{A}= 1.1–1.3, whereas, for Case S6, this ratio was D/D

_{A}= 7.5–8.0.

## 4. Summary and Conclusions

- -
- When neglecting the effect of the mean strain value on the K′ and n′ parameters and considering only the parameters of the cyclic deformation curve for ε
_{m}= 0 (symmetric loads), the ratio of the total degree of fatigue damage varied from 10% for ε_{a}= 0.2% to 3.5% for ε_{a}= 0.6%. The largest differences in the calculation of the ratio of the partial degrees of fatigue damage in relation to the reference case were observed for sequence block n_{3}, where ε_{m}= 0.4%. - -
- When assuming the independence of parameter K′ from the mean strain value, the worst calculation results in relation to the reference Case A were obtained for K′ = 587.7 MPa, where the total degree of fatigue damage was, on average, four times lower than the reference case. For these simulations, the largest calculation inaccuracy was also related to the n
_{3}block load sequence, where the mean strain value was the largest (0.4%). - -
- When considering the independence of parameter n′ from the mean strain value, the best results in terms of the degree of fatigue damage calculation were achieved for n′ = 0.1507 (obtained for a symmetric load, ε
_{m}= 0). The differences in the ratios of partial and total degrees of fatigue damage compared to the reference case were in the range of −20% to 4%. Similar results were obtained for Case B, where parameters K′ and n′ characterized the cyclic strain curve for symmetric loads. - -
- It can be concluded that the third sequence n
_{3}, where the biggest mean strain value was applied (ε_{m}= 0.4%), led to the largest inaccuracy. A higher value of mean strains, thus, increases the sensitivity of the algorithm toward applied parameters K′ and n′.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Mean strain effect on cyclic stress–strain curves: (

**a**) Changes in the coefficients K′ and n′; (

**b**) plastic region of stress–strain curves.

**Figure 7.**The ratio D

_{iB}/D

_{iA}of the partial degrees of fatigue damage and the ratio D

_{B}/D

_{A}of the total degree of fatigue damage.

**Figure 10.**Simulation results for Cases S1–S6. (

**a**) Results for Case S1, K′ = 722.6 MPa; (

**b**) Results for Case S2, K′ = 693.1 MPa; (

**c**) Results for Case S3, K′ = 587.7 MPa; (

**d**) Results for Case S4, n′ = 0.1507; (

**e**) Results for Case S5, n′ = 0.1424; (

**f**) Results for Case S6, n′ = 0.1170.

Step No. | Description of Operation | Equation |
---|---|---|

1. | Material properties: E, Young’s modulus; K′, cyclic strength coefficient; n′, cyclic strain-hardening exponent. | |

2. | Input data: ε_{A}, ε_{B}, ε_{a}, and ε_{m} | |

3. | alculation of σ_{A} for given K′ and n′ (*) | ${\epsilon}_{a}=\frac{{\sigma}_{A}}{E}+{\left(\frac{{\sigma}_{A}}{{K}^{\prime}}\right)}^{\frac{1}{{n}^{\prime}}}$ |

(*) Values of the coefficients K′ and n′ depend on the current values of the average strain ε_{m}. However, this requires additional tests to determine the functions K′ = f (ε_{m}) and n′ = f (ε_{m}). | ||

4. | alculation of ε_{Apl} | ${\epsilon}_{Apl}={\epsilon}_{A}-\frac{{\sigma}_{A}}{E}$ |

5. | alculation of σ_{a} for given K and n obtained for ε_{m} = 0 | ${\epsilon}_{a}=\frac{{\sigma}_{a}}{E}+{\left(\frac{{\sigma}_{a}}{K}\right)}^{\frac{1}{n}}$ |

6. | alculation of ε_{apl} | ${\epsilon}_{apl}={\epsilon}_{a}-\frac{{\sigma}_{a}}{E}$ |

7. | Calculation of mean stress σ_{m} | ${\sigma}_{m}=E\left({\epsilon}_{miz}+{\epsilon}_{ap}-{\epsilon}_{Apl}\right)$ i refers to the next load sequence |

8. | Resulting parameters (σ_{a}, σ_{m}) |

ε_{m} (%) | 0 | 0.2 | 0.4 |

K′ (MPa) | 722.6 | 693.1 | 587.7 |

n′ | 0.1507 | 0.1424 | 0.117 |

ε_{a} | σ_{a1}(MPa) | σ_{a2}(MPa) | σ_{m2}(MPa) | σ_{a3}(MPa) | σ_{m3}(MPa) | D_{1A} | D_{2A} | D_{3A} | D_{A} |
---|---|---|---|---|---|---|---|---|---|

0.0020 | 246 | 249 | 3 | 253 | 7 | 0.0026 | 0.0028 | 0.0032 | 0.0086 |

0.0030 | 275 | 278 | 3 | 277 | 2 | 0.0109 | 0.0117 | 0.0115 | 0.0341 |

0.0040 | 294 | 296 | 2 | 292 | −2 | 0.0258 | 0.0270 | 0.0248 | 0.0776 |

0.0050 | 308 | 309 | 1 | 303 | −5 | 0.0470 | 0.0480 | 0.0426 | 0.1376 |

0.0060 | 319 | 320 | 1 | 312 | −7 | 0.0739 | 0.0753 | 0.0650 | 0.2142 |

ε_{a} | σ_{a1}(MPa) | σ_{a2}(MPa) | σ_{m2}(MPa) | σ_{a3}(MPa) | σ_{m3}(MPa) | D_{1B} | D_{2B} | D_{3B} | D_{B} |
---|---|---|---|---|---|---|---|---|---|

0.0020 | 246 | 246 | 0 | 246 | 0 | 0.0026 | 0.0026 | 0.0026 | 0.0078 |

0.0030 | 275 | 275 | 0 | 275 | 0 | 0.0109 | 0.0109 | 0.0109 | 0.0328 |

0.0040 | 294 | 294 | 0 | 294 | 0 | 0.0258 | 0.0258 | 0.0258 | 0.0775 |

0.0050 | 308 | 308 | 0 | 308 | 0 | 0.0470 | 0.0470 | 0.0470 | 0.1411 |

0.0060 | 319 | 319 | 0 | 319 | 0 | 0.0739 | 0.0739 | 0.0739 | 0.2218 |

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**MDPI and ACS Style**

Pawliczek, R.; Lagoda, T. Investigation of Changes in Fatigue Damage Caused by Mean Load under Block Loading Conditions. *Materials* **2021**, *14*, 2738.
https://doi.org/10.3390/ma14112738

**AMA Style**

Pawliczek R, Lagoda T. Investigation of Changes in Fatigue Damage Caused by Mean Load under Block Loading Conditions. *Materials*. 2021; 14(11):2738.
https://doi.org/10.3390/ma14112738

**Chicago/Turabian Style**

Pawliczek, Roland, and Tadeusz Lagoda. 2021. "Investigation of Changes in Fatigue Damage Caused by Mean Load under Block Loading Conditions" *Materials* 14, no. 11: 2738.
https://doi.org/10.3390/ma14112738