# Residual Stress Distribution Monitoring and Rehabilitation in Ferromagnetic Steel Rods

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## Abstract

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## 1. Introduction

## 2. Stress Monitoring

#### 2.1. Permeability Monitoring

#### 2.2. MDL Stress Monitoring

#### 2.3. High Frequency Eddy Current Based Stress Monitoring

## 3. Stress Annihilation

## 4. Discussion

#### 4.1. On the Stress Monitoring and Annihilation

- All measurements are in a relatively good agreement. Stresses are detected at the same positions, which is the important parameter to be detected. Bearing this in mind, the stress annihilation can be realized by localized heating, which, from the initial evidence and performance reported in this paper, has a promising potential.
- The permeability measurements can be used to determine the actual residual stresses. This comes at the cost of the speed of monitoring: at 0.1 Hz per point, at least 1000 s are required for 100 measurements. The spatial resolution of this measurement is in the order of 10 mm, due to the length of the search coil.
- The magnetoelastic uniformity indicates the consequent decrease of the output voltage, demonstrating an accumulation of stresses in the corresponding response. However, the method is fast, requiring a few tens of seconds for 100 measurements. Since the MDL voltage output is correlated with the permeability of the tested material [28], the correlation of permeability and residual localized stresses can also be determined. However, it was out of the scope of this paper to provide such detailed information. The variation of the voltage output suffices to decide whether rehabilitation through localized heating is necessary. Future work is underway to describe in detail the amount of current and time required, involving multi-parametric finite element analysis. The spatial resolution of this measurement is in the order of 1 mm, due to the length of the search coil.
- The sound velocity monitoring indicates the actual residual stresses. The speed of measurement is much higher than the permeability measurements, requiring a few tens of seconds to perform 100 measurements. These measurements are complementary to the magnetoelastic non-uniformity measurements, providing additional proof of residual stresses. However, the spatial resolution of the sound velocity measurement is in the order of 70 mm, due to the distance required between excitation and search coils.
- The fast magnetoelastic uniformity measurement offers the indication of the change of residual stresses at consequent infinitesimal point. It is the fastest method from all, requiring a few milliseconds to monitor the whole length tested, with the highest possible accuracy. Apart from being the fastest measurement from all methods studied in this paper, it also offers sufficiently good results, offering signals illustrating residual stresses at the same position such as permeability measurements, the other magnetoelastic measurements, as well as the eddy current measurements. The spatial resolution can easily be below 0.1 mm [32], dependent on the clock of the oscillator, performing the signal processing for the A/D conversion process.
- The eddy current measurement can determine localized geometrical changes, such as those caused by the steel sphere hammering. Such measurement is fast due to the high excitation frequency. Bearing in mind that the eddy current response depends on the product of conductivity and permeability, having determined the amplitude of permeability by low (and high) frequency measurements, the conductivity variation can also be determined, which may be useful for certain applications. The spatial resolution of this measurement is in the order of 1 mm, due to the length of the search coil.
- As a result of optimum instrumentation and measurement, as well as monitoring time, the fast magnetoelastic uniformity measurement based on the MDL technique and the eddy current geometrical changes measurement can be used for the precise and complementary determination of residual stress determination. However, if it is not possible to use the long coil method required for the fast magnetoelastic measurement, then the combination of the magnetoelastic uniformity and the eddy current measurement should also be acceptable, at the expense of time monitoring.

#### 4.2. SWOT Analysis

#### 4.2.1. Strengths of the Method

#### 4.2.2. Weaknesses

#### 4.2.3. Opportunities

#### 4.2.4. Threats

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Experimental arrangement for B-H loop stress monitoring in rods: (

**a**) the schematic of the primary (1) secondary (2) coil arrangement; (

**b**) the actual set-up; (

**c**) typical waveform of the secondary coil output.

**Figure 3.**Dependence of the maximum differential permeability on the position of the primary-secondary coil, representing the effect of stress due to the hammering of the magnetic steel rod.

**Figure 4.**MDL set-ups aiming at stress distribution monitoring: (

**a**) magnetoelastic uniformity measurement, where the excitation coil (1) remains in a fixed position and the search coil (2) moves along the length of the magnetic rod, with an output dependent on the localized residual stresses; (

**b**) sound velocity measurement, where the excitation (1) and search (2) coils are fixed in distance and the whole assembly moves along the length of the rod, monitoring the change of the longitudinal sound velocity or the corresponding changes in the MDL delay time; (

**c**) fast magnetoelastic uniformity tests indicating the difference in residual stresses between consequent infinitesimal volumes, where (1) is the excitation coil and (2) the long search coil.

**Figure 5.**MDL response dependence on residual stresses: (

**a**) MDL magnetoelastic uniformity (MEU); (

**b**) sound velocity distribution; (

**c**) fast magnetoelastic uniformity measurements.

**Figure 10.**The proposed solution for point stress monitoring: (

**a**) consequent rings of Hall or MR sensors with radial shift to cover the whole area; (

**b**) pancake coils to excite and detect localized elastic pulses; (

**c**) eddy current probes to monitor localized surface non-uniformities.

**Figure 11.**The proposed localized RF induction heater: pancake coils are used to cause localized heat.

Sensor | Sensitivity of Measurement | Uncertainty of Measurement | Speed of Measurement | Ease of Measurement | Spatial Resolution |
---|---|---|---|---|---|

Permeability sensor | Able to detect 10 MPa residual stress | Certified <1% | 10 s per point | Easy: small electromechanical coil–coil arrangement | 10 mm |

Magnetoelastic uniformity sensor | Able to detect 10 MPa residual stress | Assumed to be <1% | 1 ms per point | Easy: small electromechanical coil–coil arrangement | 1 mm |

Sound velocity uniformity sensor | Able to detect 10 MPa residual stress | Assumed to be <1% | 1 ms per point | Easy: small electromechanical coil–coil arrangement | 70 mm |

Fast magnetoelastic uniformity sensor | Able to detect 10 MPa residual stress | Assumed to be <1% | 1 ms per 1000 points (integrated measurement) | Not easy: long search coil | 0.1 mm |

Eddy current sensor | Not applicable | Not applicable | 1 ms per point | Easy: small electromechanical coil–coil arrangement | 1 mm |

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**MDPI and ACS Style**

Liang, K.; Angelopoulos, S.; Ktena, A.; Bi, X.; Hristoforou, E.
Residual Stress Distribution Monitoring and Rehabilitation in Ferromagnetic Steel Rods. *Sensors* **2022**, *22*, 1491.
https://doi.org/10.3390/s22041491

**AMA Style**

Liang K, Angelopoulos S, Ktena A, Bi X, Hristoforou E.
Residual Stress Distribution Monitoring and Rehabilitation in Ferromagnetic Steel Rods. *Sensors*. 2022; 22(4):1491.
https://doi.org/10.3390/s22041491

**Chicago/Turabian Style**

Liang, Kaiming, Spyridon Angelopoulos, Aphrodite Ktena, Xiaofang Bi, and Evangelos Hristoforou.
2022. "Residual Stress Distribution Monitoring and Rehabilitation in Ferromagnetic Steel Rods" *Sensors* 22, no. 4: 1491.
https://doi.org/10.3390/s22041491