Modeling magnetization processes in steel under stress using magnetic objects

Ferromagnetic iron-based alloys used to form steel products are ubiquitous in our society, with critical applications arising in electrical steels, oil and gas pipelines, naval structures, aircraft landing gear, and automotive components.¹ Magnetic properties of these materials can be critical for such applications as electric motors and transformers, where magnetic properties have direct implications for energy efficiency. Inspection of ferromagnetic components using magnetic flux leakage (MFL)² and pulsed eddy current (PEC)³ also require the characterization of magnetic properties such as permeability and anisotropy. This enables compensation for variations of these parameters on the sizing of flaws or measurement of wall thickness, respectively. The presence of applied stress due to loading or the presence of residual stress due to manufacturing processes or plastic deformation can also lead to changes in magnetic properties, which directly affect the accuracy of inspection results obtained using MFL and magnetic Barkhausen noise.² 

The interplay between microstructure and material stress state, and its effect on the formation of domains in ferromagnetic materials, is important for the application of magnetic materials used for electrical motors⁴ and transformers,⁵ inspection of oil and gas pipelines,² naval structures,⁶ and aircraft landing gear.⁷ This interplay depends on length scales over which ferromagnetic domains form and are defined by the minimization of local magnetic energy elements of which formation of demagnetizing fields plays the largest role.¹ Domain structure forms and evolves under the applied field according to the underlying microstructure, crystallographic texture, presence of inclusions, dislocations, and material stress. Under the action of an applied magnetic field or stress, domain configurations are modified to accommodate the energy of the domain configuration.¹ 

Consideration of basic ferromagnetic energy terms at the micromagnetic scale and their extension to larger length scales has been implemented in multiscale modeling to examine coupled magnetoelastic behavior,^8,9 including hysteresis effects¹⁰ and magnetic Barkhausen noise (MBN).¹¹ While single crystals are considered as the building blocks for larger length scales,^8,11 detailed domain structure effects have not been incorporated.

Domain wall motion under tensile stress conditions has recently been investigated using magneto-optical Kerr microscopy (MOKE), which was used to show the correlation between measured domain wall velocity and MBN signal.¹² This work¹² and others¹³ have also reported the observation of domain structure refinement and its reorientation in the direction of applied tensile stress, with reverse processes occurring under compressive stress.^14,15 Examination of domain wall displacement under stress conditions using MOKE combined with magneto-optical indicator film has been used to examine the effects of cyclic stress and stability of domain wall displacement in grain oriented electrical steel.¹⁶ Recent work by Ding et al.¹⁷ has also examined domain wall dynamics under applied stress conditions, indicating that skewness of the MBN profile provides a robust feature for applied stress characterization, and have considered results in light of domain wall energy, characteristic relaxation time and distances between pinning edges.

In order to begin to quantitatively accommodate the various properties of ferromagnetic steel that affect the formation and evolution of domain structures under magnetizing conditions, a Tutorial that provides a physics based approach is presented here using the basic concept of the magnetic object (MO), first introduced by Bertotti.^18,19 The magnetic object is considered as a separate region of correlated magnetic domain structure, which in the first stage of application, is relatively independent of other neighboring domain structures. The MO is regarded as characteristically existing within single crystallographic iron grains. The MO concept has been further developed by minimizing the contributions to the total energy of the domain structure in order to explain the effects of elastic stress on magnetic Barkhausen noise signals.^1,20 In this context, the MO model has been modified from that originally proposed^18,19 by first considering a flux-closed MO under zero field conditions,^1,20,21 which can form a magnetic moment under the action of applied stress.²² The MO model is ideal in this case as it starts in the low field flux closed state and considers the exchange of magnetoelastic energy, under elastic stress conditions, with surface magnetostatic energy. This exchange results in the formation of magnetic moments within grains, which under stress gradient conditions, combine to generate a net magnetization in steel, the leakage fields of which may be detected by a magnetic field sensor.²² 

The basic magnetic object model has been used to provide an explanation for a number of observed magnetic effects in polycrystalline steel. Multiple MOs have been used to explain anisotropy in magnetic Barkhausen noise (MBN) measurements under single and dual easy axis conditions,²³ the effects of tensile stress on MBN anisotropy,²⁰ and measurement of tensor components of magnetostriction.²⁴ MOs have helped elucidate the reduction in MBN with increased temper embrittlement, as due to migration of impurity elements and carbon toward grain boundaries where they no longer directly interact with the domain structure within the MO.²⁵ The MO model has also been extended to examine how MBN changes under applied uniaxial elastic tensile stress.^25,26 Recently, MOs have been used to explain the production of self magnetic flux leakage signals under stress concentration conditions under zero applied field.²² This Tutorial presents the current state of the MO model along with prospects for its further development and application.

The reduced exchange energy associated with the separation of atoms (i.e., tensile strain) in Fe is counterbalanced by the elastic energy, which increases as a quadratic function of the strain in the crystal until equilibrium is reached at a finite strain.²⁷ The application of stress to the material will affect the state of this energy balance and, therefore, can be used to define stress dependent energy, $E σ$⁠.²⁸ The relation between the stress and strain components for a cubic crystal is given by the relation between the elastic compliance tensor and the elastic constants for a cubic crystal.²⁹ In the case of iron, $E σ$ along the crystallographic [100] direction is relevant under low and medium field conditions in the elastic stress regime. In the plastic regime, continued application of stress only produces a slight increase in interatomic spacing as planes slip past each other and dislocations are introduced, resulting in only a small change in $E σ$⁠.

Stress induced changes in magnetization reflect the underlying physical principle of the exchange of mechanical and magnetic energy. The phenomenon is also called inverse magnetostriction, the Villari effect or piezomagnetism.³⁰ Jiles and Atherton,³¹ Jiles,³² Sablik et al.,³³ Sablik,³⁴ and Shi et al.³⁵ developed a model to describe changes in magnetism with applied stress by proposing that stress can be considered equivalent to the application of a magnetic field.²² In addition to these phenomenological models³⁴ and multiscale models,^8–11 a more fundamental physics-based approach presented here is centered on the magnetic object (MO), which helps identify the effects on domain structure first within an ideal single Fe crystal incorporating various competing energy sources using energy minimization,^18,19 followed by effects that may arise within a polycrystalline matrix, consisting of many such MOs. Expressions for the total energy of the local domain structure in the MO have been developed in order to elucidate the effects of stress on magnetic Barkhausen noise.^1,20 In this regard, the MO model has been further developed by proposing a flux-closed magnetic object under zero field conditions.^1,20,21 In the absence of an applied field, the MO has been used to show the formation of a magnetic moment under the action of applied stress.²² Under zero applied field conditions a flux-closed configuration, as shown in Fig. 2, minimizes the total energy of a domain configuration. Application of stress has a number of profound effects on domain structure, and in particular, depends on the relative orientation of stress with respect to the domain structure's easy axis (defined as being parallel to the 180° domains). This includes reorientation of the domain structure's easy axis into the direction of applied uniaxial tensile stress and away from the direction of applied compressive stress.²⁰ An increase in uniaxial tensile stress leads to reorientation of the domain structure's easy axis, if not aligned with the tensile stress direction, followed by refinement of the domain structure, i.e., an increase in the number of 180° domain walls, while compressive stress has the opposite effect.²⁰ Under tensile stress conditions, the introduction of a 180° domain walls is limited by the energy cost associated with the addition of 180° domain wall area. The energy cost also depends on grain size with smaller grains permitting fewer 180° domain walls to form with stress.¹ Alternatively, the application of tensile stress will introduce energy to the MO that can go into the formation of a magnetic moment.²² The generation of many such moments can result in a bulk magnetization, thereby producing a direct physical connection between stress and induced magnetization.²² 

Sections IV–VII will examine some particular examples of where the MO formalism can be applied. The effects of stress on ferromagnetic steel domain structure, with reference to the MO model, are examined and conditions of uniaxial and orthogonal biaxial applied stress conditions are considered. The formation of different domain structures due to elastic tensile and compressive stress conditions is put forward as the mechanism for observed asymmetric response between compressive and tensile stress to the magnetization of ferromagnetic steel structures undergoing applied stress conditions as reported by Craik and Wood.³⁶ 

The interaction of the MO with a low to medium applied field motivates the examination of the change in domain structure over the entire range of magnetization.

For increasing applied field, magnetic objects can be used to qualitatively describe the progression of the domain structure up to saturation. Figure 4 shows the flux density as a function of applied magnetic field (H) for the initial magnetization curve and resultant change in domain wall configuration as represented by the MO in Fig. 2. Progression of domain structure with increasing magnetization in iron starting at the demagnetized state (i) where domain structure is flux closed; (ii) movement of primarily 180° domain walls with consequent formation of a dipole moment aligned along the easy axis direction closest to the applied field; (iii) completion of 180° domain wall motion and the end of the most rapid change in magnetization with field; and (iv) completion of 90° domain wall motion; followed by (v) domain vector rotation against the crystallographic easy axis energy into the direction of the applied field.

The movement of domain walls and therefore, growth of moments, is inhibited by the presence of pinning sites. In the MO model pinning is represented as variations in the energy landscape that can arise due to the pinning of domain walls by inclusions, impurities, and dislocations.²⁷ Pinning of domain walls due to magnetostatic coupling across martensite twins has also been identified.³⁹ The inset shown on the right side of Fig. 5 represents the movement of a domain wall, due to an applied field H, over a hypothetical energy landscape, which varies with position x. The transition from the pinning site at position 2 to the next highest pinning energy at position 3 represents a Barkhausen jump¹ with a jump distance ℓ_P, while the domain wall at position 1 remains pinned. Ding et al.¹⁷ have considered pinning edge distances and their effect on the MBN signal skewness. Changes in pinning site density via temper embrittlement and the resulting effect of pinning on abrupt changes in domain structure as measured by Barkhausen noise, including the effects of stress, have recently been considered in light of the MO model.^25,26,40

In the case, that stress is applied to the MO the energy balance of the domain structure is modified. The interaction of domain structure and stress is considered for various situations below. Here, the change in magnetoelastic energy can be compensated by a reorientation of the domain structure if it is not already aligned within 45° of the principal stress direction.²⁰ This is followed by either a refinement of the domain structure (an increase in the number of 180° domain walls),^1,20 or an increase in magnetostatic energy by the formation of a magnetic moment.²² Each of these cases will be considered separately below.

In the case of uniaxial compressive stress, for angles −45° < θ < 45° with respect to the easy axis the opposite occurs [Fig. 6(b) would be reoriented to Fig. 6(a)] since the sign in Eq. (17) becomes positive, but compressive stress now provides the negative sign. Reorientation of domain structures due to uniaxial applied stress has been reported in the literature for imaged domains by Craik and Wood³⁶ and others^{12–16,41,42} and by Barkhausen noise measurements.^20,43–46

The effect of uniaxial tensile stress on the refinement of the domain structure, or an increase in the number of 180° domain walls, which demonstrates a grain size dependence, is considered next.

Application of uniaxial tensile stress at angles θ within 45° of the easy axis results in an increase in the 180° domain volume and a decrease in the 90° domain volume with a consequent increase in the number of 180° domain walls.^46–48  Figure 7 shows this refinement of the domain structure from (a) an initial configuration with n = 2 180° domain walls to (b) with n = 3 180° domain walls. This result can be obtained from the MO model by considering the magnetoelastic and domain wall energies.^1,20

The threshold stress for the addition of a 180° domain wall increases with the number of 180° domain walls present and also depends on the size and relative dimensions of the MO. In general, the threshold stress decreases with increasing MO size. The increase in the number of 180° domain walls with tensile stress is limited since the addition of each domain wall also costs energy.^1,20 Note that under compressive stress conditions, the number of 180° domain walls that can be removed is finite, i.e., it is required that n ≥ 0, hence an asymmetry between tensile and compressive stress conditions with respect to the formation of domain structures arises. Observations of the refinement of domain structures are given by Dijkstra and Martius,⁴⁷ Craik and Wood,³⁶ Qiu et al.,⁴³ Liu et al.,¹² Perevertov et al.,^13–15 and Zeng et al.¹⁶ Barkhausen measurements on pipeline steel also support the reorientation and refinement of domain structure with applied tensile stress in steel for the typical stresses and grain sizes observed there.²⁰ 

The grain size of cold-rolled and recrystallized steel is primarily in the range of 5–20 μm,^49–51 while mild steel grains can reach 50 μm.²² A range of 5 to 50 μm is plotted in Fig. 8 for equiaxed grains (a − b = T). The equations also accommodate non-equiaxed grains that may arise under particular processing conditions.^21,40 Note that for small grain size high-strength steels the extent of domain refinement is limited, and the stress energy must be balanced by other mechanisms to be considered below. Calculated threshold stresses are consistent with those observed to have an effect on MBN response in smaller grained pipeline steel.²⁰ 

For grains in the range of sizes in steels, between 5 μm and 50 μm, moment width d as a function of stress for a cubic grain can be calculated using Eq. (29). At 250 MPa, the 5 μm size MO has two 180° domain walls (n = 2), while the 25 μm size has 5 (see Fig. 8). Equation (29) predicts an increasing moment width d with increasing elastic stress and MO (grain) size but is inversely proportional to the number n of 180° domain walls. For the 5 μm cubic grain with n = 2, stressed at 250 MPa and θ = 0° the moment width d is calculated as 0.02 μm, while for the 25 μm cubic grain with n = 5 the moment width is 0.04 μm.

Equation (35) states that stress in the elastic regime induces a local magnetization per unit volume, therefore, acting as an effective field. Note, however, that the actual direction of magnetization is doubly degenerate, meaning that magnetization may form either along the [100] direction closest to the uniaxial tensile stress direction or 180° away from it. While this can facilitate magnetization processes in the presence of a magnetic field, even the Earth's field, the energy associated with the formation of a demagnetizing field in the presence of this degeneracy will inhibit macroscopic changes in magnetization under uniform stress conditions. Pinning of domain structures will also play a role in the generation of potential leakage fields, which can also be affected by the application of stress.^52–54 

The question arises whether domain refinement or formation of magnetic moments will occur under applied tensile elastic stress conditions. It was noted above that the addition of 180° domain walls occurs as an integer step, with threshold stresses for the addition of one wall being higher and occurring less often for small grain materials than larger grained ones. Therefore, the smaller grained materials are more likely to exhibit stress induced magnetization than refinement. In addition, with the example, comparing the 5 and 25 μm cubic grain sizes above, 125 5 μm cubic grains fit into the volume of one 25 μm cubic grain. Therefore, the effect of stress induced magnetization of small grain materials is again expected to be greater.

Again, equiaxed grain sizes of 5 and 25 μm, may be compared using the moment width calculated in Sec. VI B, of 0.02 and 0.04 μm, respectively, under a uniaxial tensile stress of 250 MPa applied at θ = 0°. Using Eq. (42), this produces an equivalent energy change for the 5 μm MO of 9.4 × 10⁻¹⁶ J, requiring an equivalent field of H = 450 A/m, which is B = μ₀H = 0.57 mT. The larger 25 μm cubic grain has the energy of 1.9 × 10⁻¹⁴ J, but only produces an equivalent field of 181 A/m, which is 40% of that obtained for the 5 μm size grain. The effective field of 0.57 mT for the 5 μm size grain is 11 times larger than the nominal 0.05 mT of Earth's field.

Detection of stress induced leakage fields from lengths of steel pipe has been reported in the literature⁴¹ and was first reported by Atherton et al.^52,53 The application of stress to large structures will cause domain walls to become unpinned resulting in a change in magnetization due to stress cycling, nominally toward a lower energy state as pinning barriers are overcome. These observations may be explained by an examination of residual magnetic field under application of cyclic stress, which has shown repeatable domain wall displacements and that these are stable under cyclic stress and during relaxation time after release of stress.¹⁶ In contrast, the residual states of domain rotation are inconsistent,¹⁶ therefore suggesting that this is the mechanism that takes place during the first application of stress.

Therefore, the presence of an orthogonal stress, of the same sign as the primary stress, will reduce the effects of the primary stress on the domain structure, while if an orthogonal stress of opposite sign is introduced, the effects will be further enhanced.

The particular case of orthogonal stresses can be generalized to any biaxial stress case with the appropriate treatment of the equations presented here. The resulting expressions are, however, anticipated to be more complex than the simpler orthogonal case.

The application of MOs has been used to explain a wide variety of magnetic phenomena.^{1,12,20–22,24–26} However, as indicated here, the development is not complete. Future prospects for the MO from primarily a single object to the consideration of multiple objects and their interactions may result in the development of a more physics based hysteresis model that incorporates the effects of pinning, grain size, microstructure, texture, and elastic and plastic stress. Future work could involve the examination of the MO model in light of multiscale modeling.^8–11 Interpretation of magneto-optical Kerr microscopy (MOKE) results under stress and variable grain size may further validate the MO model presented here.

This Tutorial compiles existing work on the magnetic object model in the presence of uniaxial or orthogonal stress conditions. The magnetic object model is used to show that stress in the elastic regime can modify the domain structure as well as act as a source of effective magnetization in polycrystalline steels. Application of elastic stress modifies the existing domain structure and forms magnetic moments within grains which can generate a net magnetization within steel that, as a consequence, may exhibit leakage flux under stress gradient conditions or produce a demagnetization of steel under cyclic stress conditions. The level of local magnetization is proportional to the elastic uniaxial tensile stress and is estimated to be ten times that of the Earth's field. When orthogonal stresses of the same type arise (both tensile or both compressive), elastic stress induced magnetization is reduced or, in the case of equal stresses, will be canceled. Model predictions are consistent with experimental results that investigate changes in domain structure with stress and with magnetization measurements.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (No. RGPIN-2017-06365).

The authors have no conflicts to disclose.

The data that supports the findings of this study are available within the article.