Title Refinement of the magnetic composite model of type 304 stainless steel by considering misoriented ferromagnetic martensite particles

Refinement of the magnetic composite model of type 304 stainless steel by considering misoriented ferromagnetic martensite particles Author(s) Kinoshita, Katsuyuki Citation AIP Advances (2017), 7(5) Issue Date 2017-05-01 URL http://hdl.handle.net/2433/218242 Right © 2017 Author(s). This article is distributed under a Creative Commons Attribution (CC BY) License. Type Journal Article Textversion publisher


I. INTRODUCTION
Type 304 stainless steel (SUS304 steel) is normally paramagnetic, but it is a unique material in which plastic deformation induces a transformation to the ferromagnetic martensite phase. This characteristic has been used to assess fatigue degradation 1 and the extent of plastic deformation. 2 However, for this, it is important to develop a magnetic composite model that allows us to predict the magnetic properties of SUS304 steel. Magnetic composite models using an effective medium approach 3,4 have been developed to characterize dynamic magnetic behavior such as ferromagnetic resonance. As an alternative approach, a magnetic composite model using Eshelby's equivalent inclusion method 5 was introduced to describe the behavior of magnetoelectroelastic composite materials. 6 The latter has the advantage that it allows for coupled analysis. However, to our knowledge, a magnetic composite model to describe static magnetic properties, such as the hysteresis loop, has not been developed yet.
In a previous work, we derived a magnetic composite model 7 that combined Eshelby's equivalent inclusion method and the JilesAtherton model (JA model). 8 However, that derived model did not incorporate the orientation angle and orientation distribution of the martensite phase. In this study, we derived a magnetic composite model that allows for misoriented martensite particles, and investigated the influence of these particles on the permeability in SUS304 steel.

II. MATHEMATICAL MODEL
As shown in Fig. 1(a), needle-type particles (martensite particles, MPs) are generated in a grain of SUS304 steel as a ferromagnetic martensite phase, and those grains are distributed in a test piece. Therefore, we need to consider two different interactions: those between the MPs in a grain and those between groups of MPs. MPs in a grain are modeled as single ellipsoidal inhomogeneity (equivalent martensite particle, EMP) whose nonlinear magnetic behavior is described by the JA model. In this a Electronic mail: kinosita@energy.kyoto-u.ac.jp study, the EMPs are assumed to be prolate spheroid. The interaction between the MPs is accounted for by the interaction term α of the JA model. Based on that model, the differential magnetic susceptibility χ f of an EMP can be written as where α, k, and c are model parameters, the subscript f represents the martensite phase, H e is the effective magnetic field, M an is the anhysteretic magnetization (described by the Langevin function), δ denotes the sign of dH f /dt, and . The magnetic behavior of the composite is described by incremental equations based on the Eshelby's equivalent inclusion method, and the interaction between the EMPs by using the Mori-Tanaka theory. 9 The effect of the orientation distribution of the EMPs was introduced in the model using a method similar to those of Hatta et al. 10 and Dunn et al. 11 Let the x 3 -axis point in the longitudinal direction of the test piece and the x 3 -axis coincide with the fiber axis, as shown in Fig. 1(c). The magnetic composite is subjected to a uniform incremental magnetic fieldḢ 0 in the x 3 direction. The orientation of an EMP is defined by two angles θ and φ. Using Eshelby's equivalent inclusion method, an incremental magnetic flux in the EMP can be written in the local coordinate system through the following vector equatioṅ where µ is the differential permeability,Ḣ is the incremental average disturbance of the magnetic field (an interaction term),˙ H is the incremental magnetic field disturbed by the existence of the EMP,Ḣ * is the incremental eigen magnetic field, and the subscript m denotes the austenite phase.
The incremental eigen magnetic field is defined using˙ H and the Eshelby's tensor 10 S by Eq. (3).
Inserting Eq. (3) into Eq. (2) and rearranging the terms, we obtaiṅ Based on Fig. 1(c), a vector can be transformed between the global and local coordinates as follows: 056008-3 Katsuyuki Kinoshita AIP Advances 7, 056008 (2017) Using Eq. (6) to write Eqs. (4) and (5) in global coordinates, we obtaiṅ When the EMP orientation angles are within the ranges θ a ≤ θ ≤ θ b and φ a ≤ φ ≤ φ b , the volume average ofḢ * and˙ H over the entire composite can obtained by discrete integration: where g(θ, φ) is a probability density function, and denotes volume average. When the magnetic composite is subjected to an applied magnetic field, as shown in Fig. 1(b), the relationship between the interaction fieldḢ and ˙ H is given by 11 where V f is the martensite fraction. The relationship between the incremental magnetic flux density and the incremental magnetic field can be found by using the lawofmixtures formula 12 for the incremental magnetic flux density, together with Eqs. (2) and (11): where c denotes the magnetic composite and I is the unit matrix. The internal magnetic field of the EMP in the local coordinates, needed to calculate Eq. (1), is given aṡ For the probability density function appearing in Eqs. (9) and (10), we use the form proposed by Maekawa et al. 13 to express the fiber orientation distribution in a fiber-reinforced composite. The following equation describes the three-dimensional orientation distribution with n peaks.

III. EXPERIMENTAL METHOD
The test specimens used were cold-rolled plates of SUS304 steel (2B finish) with a gage length of 60 mm, a width of 12.5 mm, and a thickness of 2 mm. Maximum nominal strains of approximately 0.05, 0.1, 0.2, 0.3, and 0.4 were applied to them using a tensile machine and non-contact video extensometer to generate different martensite phases. Aspect ratios (major axis length/minor axis 056008-4 Katsuyuki Kinoshita AIP Advances 7, 056008 (2017) length) and orientation angles of the MPs in the test pieces were measured using image processing software, after first polishing the surface by buffing and then chemical etching. 14 A magnetic field with triangular wave form was set up along the longitudinal direction of the specimens using an electromagnet, and the magnetic flux density was measured by means of a search coil. Permeability was estimated through a differential permeability curve, obtained from the magnetic flux density curve and the evaluation parameter Figure 2 shows histograms of the measured orientation distribution of the MPs for each specimen together with the best-fit curve calculated from Eq. (14). As it was difficult to measure the three-dimensional orientation distribution of the MP, we evaluated the two-dimensional orientation distribution for φ = 0 and n = 2. Because it was difficult to etch uniformly on the entire surface of the specimen, the number of data points was not sufficient to produce a histogram. Therefore, we assumed that the orientation distribution in the range 0 • ≤ θ<90 • was the same as that in 90 • ≤ θ<180 • , and then estimated the parameters P 1 and S 1 of the probability density function as P 1 = P 2 , S 1 = S 2 . Moreover, JA parameters were calculated from the magnetization curve for SUS410S (martensitic stainless steels) because it was difficult to obtain SUS304 steel with a 100 % martensite fraction. Table I lists those JA parameters together with the obtained values of the martensite fraction, average aspect ratio, most probable orientation angle θ mod , and probability density function parameters P 1 and S 1 for each specimen. The angle θ mod decreases with increasing maximum strain up to a maximum strain of 0.2. For larger maximum strains, θ mod remains constant, but the half-width of g (θ, 0) decreases. Fig. 3(a) shows the relationship between the parameter ψ, obtained using Eq. (15) and the permeability versus magnetic field relationship of SUS304 steel calculated using the parameters in Table I and the equations in Section II, and the maximum strain. Fig. 3(b) shows the relationship between the orientation angle θ and the parameters ψ L and ψ T (where the L and T subscripts indicate that the magnetic field and permeability are parallel to the x 3 -and x 1 -axis, respectively), calculated   using the parameter values for a maximum strain of 0.2. C1, C2, and C3 indicate that the analysis was performed using θ = 0 • , θ = θ mod , and g(θ, 0), respectively. The experimental values in Fig. 3(a) are multiplied by an additional factor of 10 1 with respect to the computed values. The ψ T value for the case C1 was not plotted because it was negligible. Although the calculated and experimental values are markedly different, the qualitative behavior of ψ L is quite similar, increasing linearly with maximum strain. The quantitative difference between the experimental and analysis result is a result of estimating the J-A parameters using SUS410S. Obtaining the magnetization curve of a martensite particle is a future challenge. For a maximum strain of 0.05, the ψ L value in the case C2 is 22% smaller than in the case C1, which clearly shows that the orientation angle has effect on ψ L . On the contrary, the ψ L values in the cases C2 and C3 are about the same, even though the half-width of g(θ, 0) is different; this means that the distribution of orientation angles has little effect on ψ L . On the other hand, the effect of the orientation distribution on ψ T is observed clearly for maximum strains of 0.2 and 0.3. In particular, for a maximum strain of 0.2, the ψ T value in the case C3 is 30% larger than in the case C2. The reason for this can be understood from Fig. 3(b). If we assume that the EMPs are uniformly oriented for 0 • < θ < 40 • , we can consider that the areas under the ψ-curves in Fig. 3(b) are nearly equivalent to those in the case C3. Because ψ T is nearly zero for θ < 10 • , the area under the ψ T -curve on the interval 20 • < θ < 40 • is about seven times larger than that on 0 • < θ < 20 • . For a maximum strain of 0.2, g(θ, 0) is symmetric about θ = 20 • and has large half-width; hence, the ψ T value in the case C3 receives a significant contribution from angles 20 • < θ < 40 • and becomes larger than in the case C2. On the other hand, the area under the ψ L -curve on the interval 20 • < θ < 40 • is smaller than the area on 0 • < θ < 20 • , but only about 1.1 times. As a result, the ψ L values in the cases C3 and C2 are very similar, though the C3 value is slightly smaller.

V. CONCLUSION
In this study, we derived a magnetic composite model to account for misoriented martensite particles, and investigated the effects of these particles on the permeability curve. The developed model is able to qualitatively reproduce the change in permeability caused by misoriented martensite particles. We found that the effect of the orientation distribution on the permeability curve depends on the relative orientation between the martensite particles and the magnetic field.