Damage location and numerical simulation for steel wire under torsion based on magnetic memory method

Abstract

Torsion tests on 50 Mn steel wire were implemented and illustrated in this research to probe into the relationship between normal components of self-magnetic-flux-leakage (SMFL) signals H _p(y) and early damage of steel wire subjected to torsional stress. Besides, a modified model with due consideration of coercivity and permeability varying with equivalent stress at elastic-plastic stage was presented to simulate H _p(y) signals of steel wire under torsion accurately. Relevant numerical simulations of magneto-mechanical coupling were performed. The results show that using zero crossing point (ZCP) of H _p(y) signals to identify defect location does not apply to steel wire under torsion. However, there is crest (or trough) showing on magnetic signal curve prior to fracture. By utilizing location of crest (or trough), damage could be positioned accurately and ultimate fracture point could be forecasted in advance as well. Furthermore, simulation results obtained from the modified model are consistent with test results. Not only can study results demonstrate that MMM technique could be used for determining early damage of steel wire under torsion, but also provide certain reference for quantitative analysis about the relationship between damage and H _p(y) signals.

Keywords

Steel wire torsion magnetic memory method damage location magneto-mechanical coupling

1. Introduction

Steel wire, as the basic element of steel wire rope, has been widely used in port, shipping and numerous industries. In the case of practical engineering, performance of wire rope would be directly impacted by various technical parameters [1]. Taking hoisting as an example, greater torsional stress effect will be generated for hoisting rope if lifting height approaches kilometer depth [2]. As torsional stress of steel wire rope increases, brittle fracture will occur due to torsional performance failing to meet the needs of material toughness. Highlighting torsional performance of the material in the course of steel wire twisting into wire rope is also a must from angle of mechanical response. Therefore, it is necessary to detect and analyze the early damage of steel wire under torsion while evaluating its comprehensive performance.

In recent years, magnetic flux leakage (MFL) has been proved to be the most reliable and cost-effective measuring technique for nondestructive testing (NDT) of wire rope [3]. However, traditional MFL techniques require applying a strong artificial field on objects, which makes them incapable of determining the mechanical degradation and early damage of objects under test [4]. Compared with the traditional MFL technique, magnetic memory method (MMM) [5–7] has more technological advantages as a new-type NDT. On the one hand, the geomagnetic field instead of an applied strong artificial field is used as the stimulus source. On the other hand, it is good for assessing early damage and fully formed defect [8].

Based on MMM, magneto-mechanical effect and physical mechanism of the spontaneous magnetic phenomenon have been studied through abundant experiments [9]. Xu et al. [10] analyzed MMM test results for buried welding cracks under different tensile loads and found magnetic signals significantly affected by buried depth of crack; Bao et al. [11] demonstrated that the maximum magnetic gradient of H _p(y) signal was effective in characterizing the degree of stress concentration by tensile tests of Q235 sheet steel; Kolařík et al. [12] verified that MMM can be used for the determination of the stress-concentration-zone location and detecting material defects and imperfections under fatigue load. However, it is not clear whether the method of damage characterization obtained from uniaxial tensile (or bend) of sheet steel in previous studies can be used to detect the early damage on steel wire rope under torsional stress. And the damage of steel wire resulted from torsional stress cannot be ignored during the period of their serves. Hence, the torsion test for steel wire is required.

From another perspective, numerical simulation for magneto-mechanical coupling has been studied as one of main methods to analyze the relationship between magnetic signals and damage quantitatively. Zhong et al. [13] introduced the simulation about magnetic field changes incurred by stress concentration; Yao et al. [14] analyzed remanent magnetic field signals of the ferromagnets under plastic deformation by utilizing three-dimensional finite element method; Shi et al. [15] proposed a magneto-mechanical model about magnetic memory method and the results obtained from the proposed model were compared with experimental data. However, researches mentioned above show that the existing constitutive models cannot be used to analyze the elastoplastic magneto-mechanical issues in shear stress state directly. What’s more, almost no researcher came up with a certain method to simulate the change of magnetic field on surface of steel wire when subjected to torsional stress. Therefore, the magneto-mechanical coupling simulation of steel wire under torsion is further studied in the paper.

Three sets of steel wires torsional tests were performed to extract and analyze the H _p(y) signals on surface of specimens in different torsion cycles. In addition, taking a full account of the variation of coercivity and permeability with equivalent stress at elastic-plastic stage, a modified model of magneto-mechanical was proposed. Then, the magneto-mechanical coupling simulation of steel wire under torsion was executed, whose results were compared with experimental results.

2. The mechanism of MMM

The essence of MMM is the magneto-mechanical coupling of ferromagnetic component under weak magnetic field (like geomagnetic field).

For ferromagnetic material, the total free energy of magneto-crystalline under no stress state can be expressed by Eq. (1). $\begin{eqnarray}\displaystyle E=E_{k}+E_{ms}+E_{el}+E_{n}+E_{d} & & \displaystyle\end{eqnarray}$ (1) where E _k is magneto-crystalline anisotropy energy; E _ms is magneto-elastic energy; E _el is elastic energy; E _n is external magnetic field energy; E _d is demagnetizing energy.

When subjected to external force, the crystal of ferromagnetic component generates greater stress and strain energy. The total free energy in crystal at this point can be referred to Eq. (2). $\begin{eqnarray}\displaystyle E=E_{k}+E_{ms}+E_{el}+E_{n}+E_{d}+E_{{\sigma}} & & \displaystyle\end{eqnarray}$ (2) where E _σ is the stress energy of materials under external force.

According to Ref. [16], when the ferromagnetic component subjects to stress σ and the material is assumed as magnetostrictive isotropy, the internal stress energy E _σ can be presented in Eq. (3). $\begin{eqnarray}\displaystyle E_{{\sigma}}=-\frac{3}{2}{\sigma}{\lambda}_{{\sigma}}\cos ^{2}{\theta} & & \displaystyle\end{eqnarray}$ (3) where 𝜆_σ is magnetostrictive coefficient of material; θ is the subtended angle between stress direction and magnetization direction.

Not only can stress increase elastic energy and magneto-elastic energy of ferromagnetic component, but change magnetization direction of domain walls as per magneto-elastic effect [17] and magneto-mechanical effect [18]. With the completion of domain directional movement, such irreversible magnetization vector will still exist. Consequently, the SMFL signals on component surface will also exhibit some regularity.

Under the stress and geomagnetic field, the directional movement of domain and magnetostriction effect in ferromagnetic component will cause inconsecutive distribution of macro-magnetic properties in stress concentration zone (SCZ). This status is still retained even after the stress is released. It is called magnetic memory effect. According to pervious experimental research, two primary criteria are commonly considered in MMM tests to justify the possible defect locations: one is zero cross point (ZCP) where normal component of SMFL signal H _p(y) changes its polarity; the other one is that gradient dH _p(y)∕dx reaches a peak value (see Fig. 1) [16].

Fig. 1.

Two primary criteria of MMM technique.

3. Experiment

Steel wire specimens, as the experimental objects, came from 6 × 37 special-shaped triangular strand 50 Mn steel wire rope. The chemical components and mechanical properties could be referred to Tables 1 and 2. The specimens were grouped as A, B and C with diameter 2.750 mm, 2.350 mm and 1.670 mm, respectively. The effective torsion length is 300 mm, and the measurement length is 200 mm. Each end of the specimen was reserved 50 mm for immobilizing by grip holders. The testing line was set along lengthwise direction, which had altogether 21 measured points made with 10 mm gap between adjacent two points. The measured points and the specified dimension of specimen are shown in Fig. 2.

Fig. 2.

The specified dimension of specimen and measured points on testing lines.

Table 1

The chemical composition of tested material (wt%)

Material	C	Si	Mn	S	P	Cr	Ni	Cu
50 Mn	0.48--0.56	0.17--0.37	0.70--1.00	≤0.035	≤0.035	≤0.25	≤0.30	≤0.25

Table 2

The mechanical properties of tested material

Material	σ_s/(MPa)	σ_b/(MPa)	Elasticity modulus/(MPa)	Poisson’s ratio
50 Mn	1450	1680	1.91 × 10⁵	0.3

Three specimens in group A were numbered as A-1, A-2 and A-3, respectively. Groups B and C were similar to A. The specimens were performed with a home-made torsion testing machine (see Fig. 3), whose load error was within ±0.1%. Counter weight is 70 N. In Fig. 3, the fixture B was fixed and the fixture A rotated relative to fixture B. The torsion loading was controlled by torsional cycles. One torsional cycle means that fixture A rotates 360 degrees relative to fixture B. Meanwhile, the rotation angle on specimen cross-section is 2π. After n loading cycles, the rotation angle is 2πn. According to the difference of torsional cycles, referring to the standard steel wire ropes for important purposes (GB 8918-2006), the grades of torsion loading were divided into 0 cycle, 3 cycles, 6 cycles, 9 cycles, 19 cycles, 29 cycles, and ultimate fracture. Groups B and C were similar to A. When the specimen rotates to preset cycles, the testing machine is manually shut off and the MMM signals are tested.

Fig. 3.

Torsion testing machine.

Magnetic signals were collected online by EMS-2003 MMM apparatus after each loading (see Fig. 4), whose measuring range is ±1000 A/m. A Hall pencil-type probe with sensitivity up to 1 A/m was used during the test. In order to reduce signal interference caused by matching the apparatus to probe, the probe was normalized before testing. The end of probe was close and perpendicular to the surface of specimen in order to decrease the influence of lift-off. The magnetic signals were gathered three times in each testing point and the averages were calculated to reduce random error.

Fig. 4.

The magnetic signals testing equipment of specimen.

4. Results

The torsion tests of all specimens were performed, and the changing laws of H _p(y) signals along the direction of testing line in different torsion cycles were analyzed. During the whole loading process, the changing laws of H _p(y) signals can be divided into three stages: (1) Initial stage; (2) Loading stage; (3) Ultimate fracture.

4.1. Initial stage

As the changing laws of magnetic signals are similar in same group, specimen A-1, B-1 and C-1 are taken as example to analyze in detail.

Figure 5 presents without the applied load, the variation of H _p(y) signals on specimen surface along testing line in initial state. The curves of H _p(y) signals exhibit the phenomenon of ZCP and have some fluctuations along the direction of testing line. The irregular fluctuations of H _p(y) curves are caused by the random remanent magnetic field. During processing and manufacturing, the steel wires have subjected to multiple cold drawing and heat treatment, which resulted in remanent magnetic field being generated in specimen. Such kind of remanent magnetic field pose a certain impact on magnetic signals.

Fig. 5.

The variational curves of H _p(y) signals along testing line in initial stage.

4.2. Loading stage

Figure 6 illustrates the variation of H _p(y) signals along testing line during loading stage. One can observe that the curves of H _p(y) signals for all specimens become more regular with an increasing of loading cycles, which can be explained by piezomagnetic effect [19]. In the process of loading, the stress field in specimen will overcome the remanent magnetic field to make magnetic domains move directionally. Hence, the regularities of H _p(y) signals will appear more unified induced by the reorientation of magnetic domains. Moreover, during the loading stage from 3 cycles to 19 cycles, the changing laws of H _p(y) signals about each specimen are similar. The reason for this phenomenon is correlated with the changing laws of magnetization during elastic-plastic state.

Fig. 6.

The variational curves of H _p(y) signals along testing line in loading stage.

According to the basic assumption of elastic-plastic mechanics, the distribution of shear stress on cross section of steel wire under torsion in elastic state is shown in Fig. 7. The maximum shear stress of cross section τ_max can be expressed by $\begin{eqnarray}\displaystyle {\tau}_{\max }=\frac{2{\pi}nGR}{l} & & \displaystyle\end{eqnarray}$ (4) where n is torsional cycles; G is shear modulus; R and l are the radius and length of steel wire, respectively.

Fig. 7.

The distribution of shear stress on the cross section.

And in elastic limit state, the cross section of specimen begins to yield when shear stress τ_y and yield stress σ_y satisfies von Mises yield criteria: $\begin{eqnarray}\displaystyle {\tau}_{y}={\sigma}_{y}/\sqrt{3}. & & \displaystyle\end{eqnarray}$ (5)

Combining Eqs (4) and (5), the torsional cycles n in elastic limit state can be expressed as $\begin{eqnarray}\displaystyle n=\frac{{\sigma}_{y}l}{2\sqrt{3}{\pi}GR} & & \displaystyle\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle G=\frac{E}{2(1+{\nu})}. & & \displaystyle\end{eqnarray}$ (7)

Taking specimen A-1 in experiment as an example, the length l, yield stress σ_y and radius R of specimen are 300 mm, 1450 MPa and 2.75 mm, respectively. Shear modulus G can be calculated by Eq. (7). Therefore, it can be found by calculation that the specimen will reach elastic limit state while the loading cycle is no more than one cycle. That means the outer surface of steel wire enters plastic state after one loading cycle.

Fig. 8.

The distribution of shear stress in plastic state.

During plastic state, with the increase of torsion loading, plastic region extends inward from outer surface of specimen (see Fig. 8). Given as r _y is the boundary of elastoplastic region, shear stress τ on cross section equals $\begin{eqnarray}\displaystyle {\tau}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{r}{r_{y}}{\tau}_{y} & (\text{Elastic region}∼0\leq r<r_{y})\\[10.0pt] {\tau}_{y} & (\text{Plastic region}∼r_{y}\leq r\leq R).\end{array}\right. & & \displaystyle\end{eqnarray}$ (8)

By means of the theoretical analysis about mechanical properties of steel wire under torsion, the outer surface of specimen enters plastic state after one loading cycle, and the plastic region extends inward from outer surface of specimen with the increase of loading cycles. In fact, magnetization does not change so much after the specimen enters plastic state according to Jiles magneto-mechanical constitutive relation (see Fig. 9) [18]. Hence, magnetic memory signals will become more regular with an increasing of loading cycles and basically remain unchanged.

Fig. 9.

Magnetization calculated by Jiles models.

However, when further increasing the load to 29 cycles, the curves of H _p(y) signals will exhibit some peak points of magnetic signals. Such as specimen A-1, obviously, a trough appears at the left side of testing line and a crest appears at the right side. For specimen B-1, there are also crest and trough on curves of H _p(y) signals, but the position of crest or trough on testing line is not completely consistent with that in specimen A-1. For specimen C-1, the changing laws of H _p(y) signals are similar to specimen A-1 and B-1, but the difference is that only one crest appears on the curves of magnetic signals.

4.3. Ultimate fracture

As the changing laws of magnetic signals are similar in same group, specimen A-1, B-1 and C-1 are taken as example to analyze in detail.

Figure 10 shows the variation of H _p(y) signals along testing line after ultimate fracture. For specimen A-1, it’s obvious to see that H _p(y) signals change sharply from positive to negative values at fracture position where 150 mm and 160 mm along the direction of testing line. Moreover, the amplitude of H _p(y) signals has a remarkable increase compared with that in loading stage. Specimen B-1 and C-1 have similar phenomenon.

Fig. 10.

The variational curves of H _p(y) signals along testing line after fracture.

5. Discussion

According to the test results mentioned above, it clearly indicates that the H _p(y) signals have larger fluctuations in initial state. And after fracture, the changing amplitudes of H _p(y) signals are too large to compare with the H _p(y) signals in other stages. Therefore, the variational curves of H _p(y) signals for each specimen without considering initial state and ultimate fracture are analyzed further from Fig. 6.

During loading, the overall trend shows that the curves of H _p(y) signals change from negative to positive along testing line. However, compared with signals in the middle of specimen, it is found that the magnetic signals on both ends of specimens fluctuate drastically, especially for groups A and B. The reason for this phenomenon is that the magnetic signals at both ends of specimens have been affected by the grip holders which are made by ferromagnetic material. Plus, it should be noteworthy that not all specimens have both crest and trough. For A-2, A-3, B-3 and all specimens of group C, only one crest or trough appears on curves of H _p(y) signals. The main reason is that within the scope of measured line, only one micro-crack generates on the surface of specimen. Compared with the research of previous scholars, this experiment also shows that the position of ZCP has no direct relationship with the position of defect or SCZ for specimen. Namely, the primary criteria based on ZCP of H _p(y) signals cannot be suitable to verify the damage of steel wire under torsion.

By means of observing the location of crest or trough on curves of H _p(y) signals, and comparing with the location of fracture for different specimens, it can be determined that ultimate fracture position of specimen has a corresponding relationship with the position of crest (or trough), as shown in Table 3. And the deviations do not exceed ±5 mm.

Table 3
The comparison between the position of crest or trough and the position of fracture

Specimen number Position of peak point/(mm) Position of fracture/(mm)

Position of crest Position of trough

A-1 155--160 41--43 160

A-2 55--60 - 60

A-3 - 145--150 148

B-1 150--160 55--60 57

B-2 75--80 160--165 163

B-3 100--105 - 103

C-1 110--115 - 112

C-2 - 112--115 113

C-3 100--105 - 101

Specimen number	Position of peak point/(mm)	Position of fracture/(mm)
A-1	155--160	41--43	160
A-2	55--60	-	60
A-3	-	145--150	148
B-1	150--160	55--60	57
B-2	75--80	160--165	163
B-3	100--105	-	103
C-1	110--115	-	112
C-2	-	112--115	113
C-3	100--105	-	101

(“-” means that there is no crest or trough on magnetic signals curve of specimen.)

From the above, the potential positions of damage and ultimate fracture can be reflected by the peak positions of H _p(y) signals. For the steel wire under torsion, if the crest or trough appears on the curve of H _p(y) signals, it can be inferred that the damage occurs here and will evolve into fracture eventually. Consequently, it can be concluded that the position of crest or trough on curve of H _p(y) signals not only can be used for damage positioning before fracture, but forecasting position of ultimate fracture accurately.

6. Numerical simulation of magneto-mechanical coupling

6.1. Modified model of magneto-mechanical coupling

It is well-known that magnetic properties of ferromagnetic materials are stress-dependent [20]. For example, the coercivity H _c and permeability 𝜇 of ferromagnetic materials may be changed in the order of 100% by the stress with in the elastic limit [21]. And the change of permeability and coercivity will result in the variation of SMFL signals. Meanwhile, the influence of coercivity on magnetic signals is greater than permeability in plastic stage. Therefore, considering the effects of permeability and coercivity on magnetic field during elastic-plastic stage, a modified model of magneto-mechanical coupling was proposed.

In elastic stage, coercive field $H_{c}^{{\sigma}}$ varies linearly with stress σ applied to the sample, which satisfies Eq. (9) [22]. Meanwhile, the energy conservation model (As shown in Eq. (10)) mentioned in Ref. [23] can be used to describe the relationship between permeability 𝜇_σ and stress σ in elastic stage. $\begin{eqnarray}\displaystyle H_{c}^{{\sigma}}=k{\sigma} & & \displaystyle\end{eqnarray}$ (9) where $H_{c}^{{\sigma}}$ is the coercivity of material under stress σ; k is a coefficient related to material. For high-carbon steel, referred to experimental data in Ref. [22], the coefficient is taken as k = 3 (A/m)⋅MPa⁻¹. $\begin{eqnarray}\displaystyle {\mu}_{{\sigma}}={\mu}+\frac{3{\lambda}_{m}{\mu}_{0}{\mu}^{2}{\sigma}}{B_{m}^{2}-3{\lambda}_{m}{\mu}_{0}{\mu}{\sigma}} & & \displaystyle\end{eqnarray}$ (10) where 𝜇_σ is the relative permeability of material under stress σ; 𝜇₀ = 4π ×10⁻⁷ H/m is the permeability of vacuum; 𝜇 is the initial relative permeability; B _m and 𝜆_m are the saturation flux density and saturation magnetostriction coefficient, respectively. 𝜇, B _m and 𝜆_m are only related with the material property. In this simulation process, 𝜇 = 280, B _m = 1.9 T, 𝜆_m = 5 ×10⁻⁶.

In plastic stage, magnetic domains in ferromagnetic materials undergo irreversible reorientation and many slip bands also generate, which will make permeability change sharply. And the change of permeability will cause the increase of coercive field. According to the test results of Refs. [24,25], with the increase of strain in plastic stage, coercivity increases linearly at the rate of 0.024 (see Fig. 11), and permeability decreases linearly at the rate of 0.020 (see Fig. 12). Moreover, the relationship between stress and strain could be considered as linear because the double-linear isotropic hardening mode was used in this simulation process. Consequently, for simplifying calculation, it can be assumed that with the increase of stress in plastic stage, the permeability decreases linearly, and the coercivity increases linearly at same rate (As shown in Eqs (11) and (12)). $\begin{eqnarray}\displaystyle {\mu}_{{\sigma}}={\mu}^{\prime }\left(1-\frac{{\sigma}-{\sigma}_{y}}{{\alpha}}\right) & & \displaystyle\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle H_{c}^{{\sigma}}=H_{c}^{\prime }\left(1+\frac{{\sigma}-{\sigma}_{y}}{{\alpha}}\right) & & \displaystyle\end{eqnarray}$ (12) where 𝜇^′ and $H_{c}^{\prime }$ are the permeability and coercivity when ferromagnetic component reaches elastic ultimate state, respectively; σ_y is the yield stress of material; 𝛼 is the parameter which has the same unit with stress.

Fig. 11.

The changing rules of coercivity with strain.

Fig. 12.

The changing rules of permeability with strain.

The existing models about magneto-mechanical coupling were proposed based on uniaxial tensile stress state. However, the steel wire under torsion is mainly subjected to shear stress. Hence, it is not appropriate to apply these models directly to analyze of steel wire under torsion. To solve this problem, a uniaxial equivalent stress was introduced, which would change the magnetic behavior in a similar manner to a complex stress situation [26]. This approach follows the classical equivalent stress definitions used in mechanics such as Von Mises equivalent stress for plasticity [27]. According to Von Mises equivalent stress criterion, the relationship between equivalent stress σ_e and shear stress τ for steel wire under torsion can be expressed as follows [28] $\begin{eqnarray}\displaystyle {\sigma}_{e}\text{} & =\text{} & \displaystyle \frac{1}{\sqrt{2}}\sqrt{({\sigma}_{x}-{\sigma}_{y})^{2}+({\sigma}_{y}-{\sigma}_{z})^{2}+({\sigma}_{z}-{\sigma}_{x})^{2}+6({\tau}_{xy}+{\tau}_{yz}+{\tau}_{zx})^{2}}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \sqrt{3}{\tau}_{yz}.\end{eqnarray}$ (13)

Therefore, considering the variation of permeability and coercive force with equivalent stress, such modified model of magneto-mechanical coupling can be given as follows: $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}ll@{}}{\mu}_{{\sigma}}={\mu} & ({\sigma}_{e}=0)\\[10.0pt] \displaystyle {\mu}_{{\sigma}}={\mu}+\frac{3{\lambda}_{m}{\mu}_{0}{\mu}^{2}{\sigma}_{e}}{B_{m}^{2}-3{\lambda}_{m}{\mu}_{0}{\mu}{\sigma}_{e}} & (0<{\sigma}_{e}\leq {\sigma}_{y})\\[15.0pt] \displaystyle {\mu}_{{\sigma}}={\mu}^{\prime }\left(1-\frac{{\sigma}_{e}-{\sigma}_{y}}{{\alpha}}\right) & ({\sigma}_{y}<{\sigma}_{e})\end{array}\right. & & \displaystyle\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}ll@{}}H_{c}^{{\sigma}}=H_{c} & ({\sigma}_{e}=0)\\[10.0pt] H_{c}^{{\sigma}}=H_{c}+k{\sigma}_{e} & (0<{\sigma}_{e}\leq {\sigma}_{y})\\[10.0pt] \displaystyle H_{c}^{{\sigma}}=H_{c}^{\prime }\left(1+\frac{{\sigma}_{e}-{\sigma}_{y}}{{\alpha}}\right) & ({\sigma}_{y}<{\sigma}_{e})\end{array}\right. & & \displaystyle\end{eqnarray}$ (15) where H _c is the coercivity of specimen under no stress stage, which is only related with material property. In this simulation process, H _c = 520 A/m.

Then, the following three steps were performed to determine the parameter 𝛼.

Firstly, the experimental H _p(y) signals at x = 50 mm, x = 100 mm and x = 150 mm in different loading cycles were extracted, respectively.

Secondly, according to basic theories about magnetization of material [29], the magnetization intensity vector M of ferromagnetic material and applied magnetic field vector H are satisfied the following relation: $\begin{eqnarray}\displaystyle \mathbf{M}={\chi}_{m}\mathbf{H} & & \displaystyle\end{eqnarray}$ (16) where 𝜒_m = 𝜇_σ −1 is magnetic susceptibility. Hence, the above formula can be rewritten as Eq. (17). $\begin{eqnarray}\displaystyle \mathbf{M}=({\mu}_{{\sigma}}-1)\mathbf{H}. & & \displaystyle\end{eqnarray}$ (17)

Actually, after ferromagnetic material is magnetized, the magnetic field surrounding them will be changed. And the variation of magnetic field caused by magnetization of ferromagnetic material is H _p, which equals magnetization intensity M. The magnetic signals H _p(y) tested in experiment are the normal components of H _p, which can be expressed as follows according to Eq. (18). $\begin{eqnarray}\displaystyle H_{p}(y)=({\mu}_{{\sigma}}-1)\cdot H(y) & & \displaystyle\end{eqnarray}.$ (18)

It is assumed that the environmental magnetic field H (y) surrounding ferromagnetic material remains unchanged. In initial stage, 𝜇_σ = 𝜇 = 280, H (y) was calculated by experimental data. Therefore, the 𝜇_σ in different torsional cycles can be obtained by Eq. (18).

Thirdly, using the relationship between permeability and equivalent stress during plastic stage in Eq. (14), the fitting curve of 𝛼--σ_e can be plotted in Fig. 13.

Fig. 13.

The fitting curves of 𝛼-σ_e in different position along testing line.

By regression analysis, it can be seen that the 𝛼 increases linearly with an increasing equivalent stress σ_e (As shown in Eq. (19)). $\begin{eqnarray}\displaystyle {\alpha}=\bar{a}+\bar{b}{\sigma}_{e} & & \displaystyle\end{eqnarray}$ (19) where $\bar{a}$ and $\bar{b}$ are average values calculated by the results in Fig. 13. $\bar{a}=-1917$ , standard deviation is 99.64; $\bar{b}=∼1.38$ , standard deviation is 0.12.

6.2. Modeling calculation

Finite element analysis (FEA) based on ANSYS software and APDL subroutine includes static field analysis and static magnetic field analysis. In static field analysis, the geometry and dimension of specimen used for this simulation are same as that in experiment. And a defect was set in middle of the model when loaded 29 cycles to reflect the change of stress and magnetic signals prior to or after damage caused. Figure 14(a) shows the 3D FEM model with defect located at center of the specimen. The width and depth of defect are 0.5 mm and 0.3 mm, respectively. Finite element used for meshing the analyzed models is solid 186 and the model after meshed is shown in Fig. 14(b). Fine mesh was made within the defect zone for getting accurate results. Boundary conditions and load were applied for modeling after meshing. Finally, the equivalent stress of elements for specimen in different torsional cycles were calculated.

Fig. 14.

Specimen model.

In static magnetic field analysis, the 3D annular magnetic field model was established to produce a homogenous environmental magnetic field with the amplitude of 40 A/m, which consisted of two 1000 mm × 1000 mm × 1000 mm permanent magnets, a 500 mm × 1000 mm × 1000 mm air layer and a yoke ring. And the coercive force and relative permeability of permanent magnet were set to 54 A/m and 1, respectively. The relative permeability of air and yoke were set to 1 and 180000, respectively. Finite element used for meshing the analyzed models is solid 96. The 3D annular magnetic field model after meshed is shown in Fig. 15. The position of specimen is shown in Fig. 16. Then, the vertical boundary condition of magnetic flux was applied to the interfaces between permanent magnet and air layer.

Fig. 15.

3D annular magnetic field model after meshed.

Fig. 16.

Sketch map of specimen position.

Finally, the results of static field were imported into the analysis of static magnetic field by modified model (Eqs (14), (15)) and relevant command flows. Then, the H _p(y) signals on surface of specimen in different loading cycles were calculated and extracted. The specific process about simulation of magneto-mechanical coupling can be seen in Fig. 17.

Fig. 17.

Simulate process of magneto-mechanical coupling.

6.3. Comparison with experiment

The plastic range on cross section of specimens in different loading cycles are shown in Fig. 18. Similar to the results of above theoretical analysis, the outer surface of specimen enters plastic state after one loading cycle, and the plastic region extends inward from outer surface of specimen with the increase of loading cycles. The range of plastic region increases constantly and much larger than the range of elastic region. Different loads only change the magnitude and do not change the variation trend of equivalent stress [26]. Hence, as shown in Fig. 19, with the number of loading cycles, the values of equivalent stress increase constantly. After the damage occurs, the local stress is far greater than the nominal stress at SCZ, and the values of equivalent stress reach maximum. Correspondingly, the magnetic induction lines vary intensively around the defect. Figures 20, 21 present that the distribution of equivalent stress and magnetic induction line around the defect.

Fig. 18.

The plastic range on cross section of specimens in different loading cycles.

Fig. 19.

The relationship between the value of equivalent stress and loading cycles.

Fig. 20.

Distribution of the equivalent stress around the defect.

Fig. 21.

Distribution of the magnetic induction line around the defect.

The results of simulation about three kinds of specimens are shown in Fig. 22. Similar to the experimental results, the curves of H _p(y) signals change from negative to positive along testing line. However, the curves of H _p(y) signals are smoother compared with the experimental results in different torsional cycles. The main reason why this phenomenon occurs is that the residual stress of heterogeneous distribution exists on the surface of specimen during manufacturing process. Moreover, the measuring direction and ambient also have some influence on the H _p(y) signals in experiment. Therefore, the H _p(y) curves of experimental results fluctuate intensively compared with that of simulative results.

Fig. 22.

The FEA results of H _p(y) along testing line in different loading cycles.

The overall trends of H _p(y) signals become more regular with the increase of loading cycles. And when loading to 29 cycles, the peak of H _p(y) signals appears on the position of damage. However, the artificial damage in simulation exists differences with the random damage in experiment, and the peak values of H _p(y) signals have direct relationship with the damage degree of specimens. Hence, it is difficult to compare the values of H _p(y) signals quantitatively when loading to 29 cycles. Accordingly, in order to further justify the accuracy of simulation and the applicability of modified model, the average H _p(y) signals values of each experimental group in the case of 19 cycles were calculated, and compared with the values of simulative H _p(y) signals at corresponding position in Fig. 23.

Fig. 23.

The comparison of H _p(y) signals between FEA and experiment (EXP) in the case of 19 cycles.

In Fig. 23, the vertical axis [H _p(y)]_EXP represents when specimen is loaded to 19 cycles, the average values of H _p(y) signals in each experimental group. The horizontal axis [H _p(y)]_FEA represents the simulated values of H _p(y) signals at the corresponding position of specimen surface calculated by FEA. The solid line is zero error line, which represents that [H _p(y)]_FEA and [H _p(y)]_EXP are completely consistent. Two dotted lines are error lines whose error range is ±30%. The obvious conclusion can be obtained that except for the individual endpoints, most of measured points are in the error range of ±30%. The main reason of this phenomenon may be that the H _p(y) signals of measured points at the both ends of specimen in experiment can be disturbed easily by grip holders. Hence, such deviations between experimental results and simulated results occur at ends of specimen. In brief, without considering the influence of grip holders at both ends, most of results calculated by FEA are relatively accurate, and the accuracy of simulation is also acceptable.

7. Conclusion

In this paper, the torsion tests about steel wire were performed and the H _p(y) signals on surface of specimens were measured synchronously. In addition, a modified model of magneto-mechanical coupling which considered the change of permeability and coercivity with equivalent stress in elastic-plastic stage was proposed for accurate simulation of magnetic field about steel wire under torsion. The following conclusions can be drawn from this work:

The position of crest or trough on curves of H _p(y) signals can be used to locate the damage on surface of steel wire under torsion and pre-judge accurately the position of ultimate fracture. That indicates magnetic memory method can be applied to detect the early damage of steel wire under torsion in practical engineering applications.

Comparing with experimental data, it can be verified that the modified model about magneto-mechanical coupling can be capable to analyze the change of leakage magnetic field about steel wire under torsion accurately. It also provides certain reference for quantitative analysis about the relationship between damage and H _p(y) signals.

The zero cross point (ZCP) on curves of H _p(y) signals, as the primary criteria of justifying defects locations, has no direct relationship with the position of defect or SCZ for specimen. It cannot be suitable for detecting the damages of steel wire under torsion.

Footnotes

Acknowledgements

The authors are grateful for the financial support received from the National Nature Science Foundation of China (Grant Nos. 51878548 and 51578449), the Key Project of Natural Science Foundation Research Plan of Shaanxi Province (Grant No. 2018JZ5013).

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