The application of ferromagnetic steel products is pervasive in society, with important applications arising in electrical steel, oil and gas pipelines, transportation infrastructure, naval structures, aircraft landing gear, and automotive components. Magnetic properties of electrical steel materials play a key role in electrical motors and transformers, with a direct impact on energy efficiency. Measurement of response to magnetization has implications for non-destructive inspection methods, such as magnetic flux leakage, magnetic Barkhausen noise, and metal magnetic memory method. Examples include flaw detection, characterization of material properties, and identification of stress state in steel. An understanding of the magnetic response of steel materials can be facilitated by the use of magnetic objects (MOs). MOs are defined as regions of relatively independent magnetic behavior, typically about the size of a grain, to which fundamental magnetic energy considerations may be applied. This Tutorial outlines mechanisms by which MOs may be applied for modeling magnetic response in steel and presents examples of their application. MOs incorporate material physical properties such as microstructure, grain size, crystallographic texture, the presence of dislocations and impurity elements, and the presence of residual stress and stress load on the component. They can also accommodate a description of the evolution of magnetic domain structure under magnetizing conditions. As the MO model incorporates fundamental physics principles, it allows estimates of physical parameters that can be used to provide insights into the connections between magnetic properties and material properties, including hardness, embrittlement, and the presence of applied and residual stress. Practical applications include non-destructive characterization of the stress state of steel and an improved understanding of magnetic processes in electrical steel. Examples where such models may be applied include magnetic Barkhausen noise and magnetic memory method for the characterization of steel materials. This Tutorial summarizes recent advances in the MO model and its applications, providing the foundation for its further development. Magnetic objects have the potential to provide fundamental explanations and could form the basis for magnetic measurements and magnetization processes, including magnetic flux leakage, magnetic Barkhausen noise, and magnetic hysteresis.

## I. INTRODUCTION

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 motors^{4} 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 effects^{10} 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 work^{12} and others^{13} 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.

## II. EFFECT OF ATOM-TO-ATOM SPACING ON FERROMAGNETIC PROPERTIES OF IRON

^{27}

^{,}$ E e x$, arising between neighboring atoms at lattice sites i and j, is given as

^{27,28}

*J*, which results in a minimum exchange energy as follows from Eq. (1). Since Eq. (1) depends on the distance between spins, $| r i \u2212 r j|$, atomic separation also affects the exchange energy. In the case of Fe, the exchange force initially increases with the distance between atoms, or in other words with increasing strain. The variation in exchange energy with atomic separation is described by the estimated Bethe–Slater curve plotted as exchange energy as a function of the ratio of interatomic distance,

*a*, to the radius

*r*of the 3d electron shell in various materials shown in Fig. 1.

^{27}

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 \sigma $.^{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 \sigma $ 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 \sigma $.

## III. STRESS INDUCED CHANGES IN MAGNETIZATION

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 models^{34} 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}

## IV. DOMAIN STRUCTURE

_{σ}), magneto-static $( E m s)$, domain wall $( E \gamma )$,

^{1}applied field $( E H)$,

^{27}and pole $( E p)$

^{37}energies written as

^{27,28}$ E K$ is constant in the MO model in the low to medium field regime, since magnetization lies along one of the crystallographic [100] easy axis directions in iron. Eliminating the constant terms, the total energy of the idealized geometry of the MO, shown in Fig. 2, within which domains are separated by 180° and 90° domain walls (at 45° to [100] direction

^{27}), with energies per unit area of $ E \gamma 180$ and $ E \gamma 90$, respectively, becomes

^{27,28}

^{28}

*N*is a form factor that relates the energy outside the object to the energy inside the object,

*V*is the uncompensated domain volume of width d shown in Fig. 3, and $ M s$ is the domain saturation magnetization. The progressive introduction of 180° domain walls, which minimizes the magnetostatic energy, reduces the total energy of the domain structure but cannot proceed indefinitely, since the domain walls themselves introduce additional energy. For a single iron crystal, the crystallographic anisotropy energy allows the formation of 90° domain walls, which arise at 45° to the [100] directions.

^{27}Under this condition and with no applied field, zero stress state, and no domain wall pinning, the lowest energy state of the MO is one of complete flux closure. A potential domain configuration shown in Fig. 2, of an equiaxed single iron crystal, contains n = 2, 180° domain walls. This will be taken as an example of the MO model, as it contains the basic elements needed to demonstrate properties of MOs including 90° and 180° domain wall elements. The 180° domain walls are orientated along the [100] direction and define the MO's easy axis of magnetization, while the 90° domain walls provide flux closure in the zero applied field case. The evolution of this domain structure under an applied field and the subsequent modification of additional energy terms, as described by Eq. (3), are considered next.

^{37}an example of which is shown in Fig. 3, and is due to the torque on an atomic dipole moment in the presence of an applied magnetic field.

^{27,28,37}The pole energy for a single dipole within a magnetic field is given by

^{27}

*m*is the magnetic moment,

*H*is the field acting at an angle

*θ*to the moment, as shown in Fig. 3, and

*ϕ*is the direction of the easy axis of the MO. The growth of the moment due to an increase in the external magnetic field is constrained by the crystallographic and exchange energies. To maintain minimum energy, domains most closely aligned with the applied field grow at the expense of neighboring domains by domain wall motion, as shown in Fig. 3. The new domain configuration causes flux lines to escape the boundaries of the MO. Here, the crystallographic anisotropy energy $ E k$ is again nearly zero, since the spins align with the crystallographic directions of the grain. However, the magnetostatic energy $ E m s$ is not zero since the MO now has a net magnetic moment. The magnetostatic energy $ E m s$ is taken to be of the same form as Eq. (5) and is given as

*T*is the MO thickness shown in Fig. 2, $ M s$ is the saturation magnetization, and

*d*is the width of the moment-producing region of the MO as shown in Fig. 3. The magnitude of the net moment of the MO is $ \mu 0 E m s$. The pole energy, which is responsible for magnetizing the MO, is derived from Eq. (6) and is given as

*a*, $d $, and

*T*are the length, width and thickness, respectively, of the moment producing region of the MO as shown in Fig. 3, and $\theta $ is the angle between the applied field

*H*and the MO magnetic moment direction (ϕ=0).

*d*of the moment-producing region may then be found by taking the derivative of the total energy and setting it to zero,

^{38}

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.

## V. PINNING

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 jump^{1} 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}

## VI. MAGNETIC OBJECTS UNDER APPLIED STRESS

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.

*V*) is given as

^{28}

^{,}

^{1,20}

*V*

_{180°}is the volume of the 180° domains and

*V*is the volume of the 90° domains. For a generalized MO domain structure with n 180° domain walls (assuming

_{90°}*a*≥

*b*), i.e., the 180° domain volume is given as

^{1,20}

^{1,20,21}

^{10}that for stresses applied at angles, θ′, for 135° < θ′ < 45° with respect to the easy axis, a reorientation of the domain configuration as shown in the transition from Fig. 6(a) and 6(b) for uniaxial tensile stress will occur. The change in energy ΔE from the angle of applied stress relative to the original easy axis direction, θ′ = θ + 90°, to the reoriented domain structure with angle θ may be obtained. Using the identity $ cos2\theta = co s 2\theta \u2212 si n 2\theta $, applied to Eq. (12) for θ and θ = θ′ − 90° the difference in energy is given by

^{20}

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 Wood^{36} 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.

### A. Domain wall refinement

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}

*n*of 180° domain walls within an MO is given by

^{1,20}

^{−3}J/m

^{2}in iron

^{18}is the energy per unit area and A

_{180°}is the surface area of the 180° domain walls within an MO. Domain wall energy for 90° domain walls for cubic domain structures is given by

^{22}

^{,}

^{−3}J/m

^{2}is energy per unit area

^{28}and $ A 90 \xb0$ is the surface area of 90° domain walls. Note that the 90° domain wall area is independent of

*n*the number of 180° domain walls. An applied tensile stress increases the exchange energy. However, for typical stresses this will not modify the direction of the easy axis that is along one of the three crystallographic easy axes.

^{27,32}In steel, the stress induced energy only equals the crystallographic energy at 4550 MPa,

^{27}which is far beyond the yield strength of even the highest strength steels considered here. However, the 180° domain wall energy is modified by the applied stress and can be obtained by considering the difference in stress induced energy on either side of the domain wall as given by Eq. (11).

^{27}As a function of angle θ relative to the easy axis this is obtained as

^{20}

^{,}

^{27}The stress dependence of the energy of the 90° domain wall may be considered negligible.

^{27}In effect the refinement of the domain structure occurs, since it is energetically favorable to reduce the volume of 90° domains perpendicular to the applied tensile stress direction.

_{σ}, as given by Eq. (15), the domain wall energies $ E \gamma 180 \xb0$ and $ E \gamma 90 \xb0$ as described by Eqs. (18) and (19), respectively, and the increase of the 180° domain wall energy with tensile stress, Eq. (20). The total energy of the MO under uniaxial tensile stress, within ±45° of the easy axis, becomes

^{1,20}

^{10}

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 *μ*m,^{49–51} while mild steel grains can reach 50 *μ*m.^{22} A range of 5 to 50 *μ*m 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}

### B. Formation of magnetic moments under uniaxial tensile stress

*d*as shown in Fig. 9. The increase in magnetostatic energy in the small moment limit

*d*<<

*a*(the field produced by the free pole of one surface makes an insignificant contribution to the field produced by the pole on the other surface), is given by

^{28}

*d*, is a reduction in the horizontal dimension of the flux-closed portion in Fig. 9 from width b to width (b–d). This modifies Eqs. (13)–(15) by substitution of b with (b–d). The moment of width d that forms is also subject to a change in energy by the applied stress as per Eq. (11), with Vol. =

*adT*. Combined with Eq. (15), the total energy of the MO, as expressed by Eq. (3) for the configuration in Fig. 9 under the applied stress condition without an applied field becomes

^{22}

*d*in Eq. (16) and using the trigonometric identity 2 cos 2θ = 1 + cos 2θ, terms containing the product

*a*×

*d*cancel, yielding a form of Eq. (25) as

^{22}

*d*, with the constraint that θ is within 45°of the easy axis, gives

^{22}

^{,}

^{28}) a good approximation for the moment width d can be given as

^{22}

^{−5}MPa

^{−1}. Note that the moment width

*d*is proportional to the applied elastic tensile stress and is also a function of the grain size as indicated by the width

*b*, the domain structure, as characterized by the number

*n*of 180° domain walls, and the angle θ of the stress with respect to the easy axis. If compressive stress is present, the moment is anticipated to form along the easy axis that is closest to the perpendicular with respect to the stress direction.

For grains in the range of sizes in steels, between 5 *μ*m and 50 *μ*m, moment width *d* as a function of stress for a cubic grain can be calculated using Eq. (29). At 250 MPa, the 5 *μ*m size MO has two 180° domain walls (*n *= 2), while the 25 *μ*m 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 *μ*m cubic grain with *n *= 2, stressed at 250 MPa and θ = 0° the moment width *d* is calculated as 0.02 *μ*m, while for the 25 *μ*m cubic grain with *n *= 5 the moment width is 0.04 *μ*m.

*V*=

*adT*, may be multiplied by the saturation magnetization, $ M S$, giving the moment in the ith MO within a structure as

^{22}

*n*is the number of 180° domain walls in the MO and $ \sigma i$ is the elastic vector stress component at the ith MO. Note that the total moment $ m$ is now a vector whose direction is primarily determined by that of the existing tensile stress

_{i}**σ**. The volume of the ith MO is given by $ V i= T i b i a i$. For a uniaxial tensile or compressive stress, without stress concentrations, the stress is in a constant vector direction. Assuming an average distribution for $ \theta i$ and average number

*n*of 180° domain walls an expression for the total stress induced moment can be written as

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 *μ*m cubic grain sizes above, 125 5 *μ*m cubic grains fit into the volume of one 25 *μ*m cubic grain. Therefore, the effect of stress induced magnetization of small grain materials is again expected to be greater.

### C. Comparison of stress and applied field

^{28}

*θ″*is measured with respect to the [110] direction as

^{18}

*θ″ =*45°

*θ*. The work done to move a single 180° domain wall a distance

*d*, and therefore, the increase in energy within the MO, is given by

*n*≥1 and the small change in length of the 180° and 90° domain walls has been neglected, as well as the potential presence of any pinning. The total work done by the field is the sum of the work done on both 180° and 90° domain walls by the applied field

Again, equiaxed grain sizes of 5 and 25 *μ*m, may be compared using the moment width calculated in Sec. VI B, of 0.02 and 0.04 *μ*m, 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 *μ*m MO of 9.4 × 10^{−16} J, requiring an equivalent field of H = 450 A/m, which is B = *μ*_{0}H = 0.57 mT. The larger 25 *μ*m 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 *μ*m size grain. The effective field of 0.57 mT for the 5 *μ*m 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 literature^{41} 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.

### D. The case of orthogonal biaxial stresses

^{55}is considered here. With the addition of a second orthogonal stress, $ \sigma 2$, in Eq. (12), with reference to Fig. 10, an additional angular dependent term is introduced at 90° to the first as

_{2}and the identity $ co s 2\theta = si n 2( 90 \xb0 + \theta ) $ was used. Considering the particular case where $ \sigma 1> \sigma 2$, domain refinement may arise as well as additional magnetostatic energy. Considering refinement first, substituting Eqs. (13) and (14) into Eq. (43) gives

*n*to

*n*+1 transition, substituting

*n*+1 into (38) and subtracting, this becomes

*n*to

*n*+1 transition under the condition that δ <<

*b*, (δ is 80 nm, the thickness of a 180° domain wall.

^{27}This is shown by the expression for the threshold stress obtained by setting $ \Delta E T=0$ and $ \sigma 1= \sigma T$ in Eq. (47) as

*n*, and therefore, no increase in the number of 180° domain walls. From Eq. (48), the application of an orthogonal tensile stress $ \sigma 2$ to an already existing tensile stress condition $ \sigma 1$ tends to reduce the effect of the original stress on the resulting domain structure, and thereby, increases the threshold stress to refine the existing domain structure. A compressive stress $ \sigma 2$ would have the opposite effect, reducing the threshold stress, leading to greater refinement. This effect on domain structure under biaxial stress conditions is supported by magnetic Barkhausen noise measurements as reported by Vengrinovich

*et al.*

^{56}and biaxial stress effects in magnetic flux leakage measurements as reported by Crouch

*et al.*

^{57}

### E. Magnetic moments under orthogonal biaxial stress conditions

*d*is considered next. Combined with Eq. (45), the total energy of the MO as expressed by Eq. (3) for the configuration in Fig. 11 under applied stress condition with a

*b*to

*(b–d)*substitution gives

*ET*/∂

*d*= 0, the width of the moment d that minimizes the energy becomes

_{S }= 2.15 A/m and $ \lambda 100$ = 2.07 × 10

^{−5}for iron

^{28}) a good approximation for the moment width for

*d*can be given as

*C*= 3.11 × 10

^{−5}MPa

^{−1}and

*d*is now proportional to the difference in orthogonal stresses. Equation (54) returns to Eq. (29) if $ \sigma 2=0$. If $ \sigma 1= \sigma 2$,

*d*becomes zero and no moment is formed. The local moment per unit volume or magnetization, M, can be obtained in the same manner as that used to obtain Eq. (35) as

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.

## VII. FUTURE PROSPECTS

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.

## VIII. SUMMARY

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.

## ACKNOWLEDGMENTS

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

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

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

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