Residual Stresses: Measurement using Magnetoelastic Effects

Residual Stresses: Measurement using Magnetoelastic Effects Magnetoelastic methods have the advantage of being rapid, entirely nondestructive, and applicable for in situ measurement on industrial plant and components. Nevertheless their high sensitivity to the internal stress is complicated and historically not well understood. In addition, they are sensitive to other factors such as material microstructure often making it difficult to interpret data. However, in the last decade of the twentieth century, significant advances were made in understanding the physical principles making this approach for practical measurement increasingly desirable. There are now several portable systems in the marketplace that have taken advantage of developments in the field and this trend is likely to increase. This article reviews fundamentals, techniques, materials, and practical issues and finishes with a few example applications. 1. Fundamentals The magnetic properties of ferromagnetic materials are sensitive to internal stress. Interaction between the atomic moments in the lattice causes alignment into magnetic domains (ferromagnetism, see Itinerant Electron Systems : Magnetism (Ferromagnetism)), which in turn generates a small strain in the lattice (magnetostriction). With internal stress, it becomes energetically favorable for the orientation distribution of magnetic domains to change, so that in steel (positive magnetostriction), the magnetization vector points towards tensile axes to reduce the magnetoelastic energy. However, this redistribution of domains tends to increase magnetostatic energy. Therefore, in general, these energy terms must balance. The changes in magnetic domain distribution mean that magnetic hysteresis, permeability, and remanence are all functions of the stress tensor. The minimum free energy (F ) for a polycrystalline material can be evaluated and the domain distribution determined as a function of the triaxial stress field. Most magnetic properties can then be predicted from this distribution. In general: F l EHjESjEKjEσjEejE

(1) ! where EH and Es are the energy of magnetization in an applied field and the self energy of magnetization:

& M.H dν E lk" µ & M.H dν # ! EH lkµ s

!

s

(2) (3)

where Hs is the field of the surface and volume pole densities and M the magnetization. EK is the magnetocrystalline anisotropy energy, which for a single cubic

crystal is given in terms of direction cosines, αi, and K , ! K , and K , the magnetic anisotropy constants: " # EK l K jK (α#α#jα#α#jα#α#)jK α#α#α# (4) ! " " # # $ $ " # " # $ Eσ is the magnetostrictive energy given in terms of the saturation magnetostriction constants, λ and λ : "!! """ Eσ lk$ λ [σ (α#k1\3) # "!! "" " jσ (α#k1\3)jσ (α#k1\3)] $$ $ ## # k3λ (σ α α jσ α α jσ α α ) (5) """ "# " # #$ # $ $" $ " Ee is the exchange free energy, given approximately by the expression: Ee l

&

2JeS# [(]α )#j(]α )#j(]α )#] dν " # $ a

(6)

where Je is the exchange integral, S the atomic spin, and a the lattice spacing. Finally, the term E repre! sents additional terms due to inhomogeneities, domain wall energy, and irreversible wall movements. At minimum energy, Eqn. (4) shows that the magnetization will align with the crystalline directions (the magnetic easy axes). Magnetization generally occurs by movement of the domain walls. Only at high applied fields will there be significant rotation away from the easy axes. Equation (5) indicates why the relative volume fraction of domains aligned along each easy axes depends upon the internal stress. In polycrystalline material with no texture, the direction cosines in the above equations must be averaged. The deviatoric stress components, si, should be substituted into Eqn. (5), the hydrostatic stress, σ , not ! contributing to energy changes, where: si l σikσ for principal axes i l 1, 2, 3 ! σ jσ jσ # $ and σ l " ! 3

(7) (8)

Calculations using this formal approach are complex for the general case. As a much simplified example, if the strain sensitivity of Eqn. (6) is ignored and the material is assumed to have no texture, the unit volume magnetoelastic and magnetostatic energy contribution from magnetic domains, the magnetization of which is closest to i, fE(i)g, can be evaluated at low applied fields (where the domains are assumed to be aligned along their easy axes, i.e., no rotation) as: fE(i)g lk$ λ (0.70σij0.15σjj0.15σkkσ ) ! # "!! (9) k0.83M Hi & with similar relationships for fE(j)g and fE(k)g with the indices cycled. By assuming that the energy distribution of magnetic domains follows a classical Boltzmann distribution and integrating over all orientations the total energy can be evaluated as a function 1

Residual Stresses: Measurement using Magnetoelastic Effects

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Figure 1 (a) Experimental and (b) theoretical DEP 1 shown as a contour level for applied biaxial stress for 30 mm rolled mild steel plate. (DEP 2 similar with stress axes reversed.)

of the triaxial stress and applied magnetic field. The free energy can then be determined by summing the energy contributions (Eqn. (9)) with the appropriate domain volume fractions. Finally the magnetic flux, B, can be determined from differentiating the free energy with respect to the applied field, H, and the low field tensoral permeability, µ(σ) l dB\dH, can be deduced. Similar approaches can be used to model anisotropic materials, higher applied fields that include domain rotation, etc. (Hauser 1994, Sablik et al. 1994). Another aspect not explicitly indicated above is that in general the applied field, H, and induced magnetic flux, B, will not be parallel because the magnetic 2

permeability is anisotropic. This has important consequences for some practical measurement methods (Sect. 2). Magnetic materials exhibit hysteresis, where the magnetization lags behind the applied field. In consequence the minimum energy condition is only realized if the material has been demagnetized. Hysteresis is caused by pinning of the magnetic domain walls; the activation energy required to unpin them means that the size and distribution of domains depend upon the magnetic history. This pinning is controlled by the microstructure, where features such as inclusions and other obstacles have greatest influence when: $ their magnetic properties are most different from the matrix, e.g., nonmagnetic; and $ their sizes are comparable with the domain wall width. Materials such as low carbon and low alloy steels as opposed to tool and other high carbon steels, exhibit relatively low hysteresis and so free energy model predictions where irreversible terms have been ignored agree well with experiment (Sect. 2). Practical measurements are usually limited to the surface of a component because field penetration into the bulk is limited by eddy current screening. If the magnet size is large compared with the field penetration, then a simple exponential formula can be derived from Maxwell’s equations for the sampling sensitivity, s(z), as a function of depth z, which defines a skin depth δ thus: s(z) l s e−zNxµ!µrkf l s e−z/δ (10) ! ! where k is the electrical conductivity and µr is the relative permeability, here assumed constant. Thus, by varying the drive and detected frequency, f, the measurement penetration can be adjusted. This proves to be a useful tool for investigating stress depth profiles.

2. Magnetic Parameters 2.1 Hysteresis Loop and Related Parameters It follows from the above that the magnetic hysteresis loop is distorted as a consequence of stress so that there will be changes in the coercivity (HB = ), rem! anence (BH = ) and permeability (dB\dH). However ! full loop measurement on plant components is usually not practical because it is not possible to generate uniform fields, and demagnetizing fields are usually present. Nevertheless, practical approaches have been developed using low field magnetic susceptibility measurement for uniaxial stress magnitude and direction measurement. Coercivity values can also be deduced from practical components because at zero magnetic flux there is no flux leakage. Therefore these measurements are independent of probe lift-off and demagnetization effects. However its sensitivity to

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stress is much lower than to microstructure, such as grain size or carbon distribution. Coercivity is strongly linked with irreversible wall movements caused by pinning points.

Measurement of incremental permeability, µ∆, can be achieved by superimposing a small amplitude modulation upon a low frequency drive field. The frequency of the secondary field must be high compared to the primary field, but still low enough to maximize penetration (Eqn. (10)). A simplified method known as directional effective permeability (DEP), has been developed for use in the field (Allen et al. 2000). This uses a fixed amplitude AC field electromagnet with equipment for sensitive impedance measurement. The induced voltage, V(t), across a coil sensor is sensitive to the average permeability along the measuring direction, µmaterial, which can be evaluated using simple magnetic circuit theory (assuming no flux loss from the electromagnet) as: µ N#Iω cos ωt ! lcore 2l l j airgapj polegap µcore xy xy µmaterial xδ

V(t) lk

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where a drive current of I sin ωt is applied to a coil of N turns wound on a C core with pole cross-section xy, relative permeability µcore and lengths l. In Eqn. (11) resistive components have been ignored. Magnetic field strengths much less than saturation are used so that the permeability, µmaterial, is assumed to be independent of the applied field. µmaterial can then be replaced by a suitable triaxial stress formulation as outlined earlier to give the predicted stress sensitivity of the DEP method. By rotating the probe, the DEP measured as a 3

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function of probe angle will be constant so that no magnetic anisotropy is measured in a stress free state in the absence of material texture. In a biaxial stress field, DEP will change sinusoidally as a function of the measuring angle, with maximum and minimum DEP along maximum and minimum stress axes respectively. This enables a measure of the two principal surface stress values and directions to be attempted. Figure 1 shows a comparison between experimental and theoretically derived biaxial response for DEP measured with the magnetic field parallel to the longitudinal axis (DEP 1) in a mild steel plate. A similar result is obtained for the transverse field orientation (DEP 2). Some workers prefer to use nonlinear harmonic analysis (NLHA) rather than a direct measure of the impedance or induced signal voltage.

2.2 Anisotropy Methods The partial alignment of magnetic domains towards the maximum positive stress axis will increase the magnetic permeability because a smaller applied magnetic field is then required to fully align the domains. Conversely the magnetic permeability measured along the minimum or most negative stress axis will be reduced. This induced magnetic anisotropy can be accurately measured, and hence the stress parameter, σ kσ (or shear for biaxial stress), determined in the " #of the material surface in two ways. plane The magnetic anisotropy system (MAS) uses a double C core electromagnet with the two sets of poles 4

aligned orthogonally. The electromagnetic voltage signal induced across a coil wound on one of the cores while driving a coil wound on the second core is proportional to the anisotropy. In the second approach, known as the stress-induced magnetic anisotropy (SMA) technique, the second core is replaced by a simple air coil. Magnetic anisotropy induced by stress results in the rotation of an induced magnetic field away from the direction in which it was applied. For example, if a magnetic field is applied at some angle other than parallel or orthogonal to a uniaxial tensile stress in steel, the induced field will rotate towards the tensile axis. The sensor alignment will link any flux in the direction perpendicular to the applied field in the plane of the component surface. Thus, as the probe assembly is rotated through 360 m on the material surface, the induced voltage will vary periodically because the rotation of flux will depend upon the angular relationship between the applied field and principal axes. The principal stress directions can be determined from angles where there is a null signal. The signal peak-topeak amplitude is approximately proportional to the difference between the two principal biaxial stresses. Theoretical development has yielded an exact relationship between the signal and the stress anisotropy so that the technique can be used for quantitative stress evaluation. Magnetic anisotropy techniques are very sensitive to stress and are almost ‘null’ methods having only a weak sensitivity to rolling texture. For rolled steel sheet or plate, where the magnetic properties are

Elastic strain (10–3)

S parameter (plastic damage)

Residual Stresses: Measurement using Magnetoelastic Effects

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Figure 5 Residual elastic strain and plastic damage measured by SMA and the positron annihilation technique across side face of plastically bent 1 m bar of 100 mm cross-section.

aligned parallel and perpendicular to the rolling direction and the permeability anisotropy at zero stress is: Stress (MPa)

µ kµ ## ∆µ l "" µ jµ "" ##

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Figure 6 MAE and SMA stress measurements compared with holehole method made on 25 mm, 0.5 m square mild steel plate containing a single-V manual metal arc weld.

anisotropic in unstressed material, assuming a simple linear relationship between permeability and stress, the following is derived: SMA(σ ,σ ) l "" ## 2∆µj∆µ(AjB)(σ jσ ) "" ## C j(AkB)(σ kσ ) "" ## 2N(1j∆µ)(1k∆µ)j2j(AjB) (σ jσ )j∆µ(AkB)(σ kσ ) "" ## "" ##

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where A, B, and C depend upon the material. Figure 2 shows experimental and theoretical data for 8 mm rolled mild steel as a function of biaxial stress. Note that even this simple empirical model predicts most of the experimentally observed biaxial response. A similar relationship to Eqn. (12) was derived when the rolling axis was not aligned with a principal stress axis. This approach has been used to derive calibration maps for a range of materials by carrying out a simple loading test with strain gauges and evaluating coefficients A, B, C, and ∆µ (small or zero for many steels). By using the free energy model in place of this linear approximation a more physical model can also be derived.

2.3 Micromagnetic Methods An early practical application for stress monitoring employed Barkhausen emission (BE). These electromagnetic pulses (or Barkhausen noise) arise from irreversible domain wall movements under an applied magnetic field, and depend upon the domain distribution and hence the stress state. The domain wall pinning sites are typically dislocations, second phases, or grain boundaries. Consequently this technique is particularly sensitive to the microstructure and mech5

Residual Stresses: Measurement using Magnetoelastic Effects

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Figure 7 Biaxial stress levels measured with MAPS through 12 mm rolled plate compared to x-ray diffraction near edge (100 mm) and center (500 mm) of 1000 mm wide plate.

anical properties of the material. Correlation of BE behavior with basic magnetic properties is difficult and the high signal frequencies (1–100 kHz) limits penetration resulting in a strong bias towards near surface layers and surface condition ( 300 µm). Nevertheless the technique has been investigated for stress measurement for several decades (Buttle et al. 1987a, Jiles 1988). The irreversible domain wall movement during the magnetization cycle will also generate elastic waves if the local direction of magnetostrictive strain is changed (i.e., by 90m walls in steel but not the more numerous 180m walls). This magnetoacoustic emission (MAE) can be detected using piezoelectric transducers with signal bandwidths 10 kHz to 1 MHz, although acoustic noise can be a problem in an industrial environment. Unlike BE, the penetration for MAE is limited only by the frequency of the applied magnetic field. Like BE, MAE measurements as a function of probe orientation may be used to determine principal stress axes and, with calibration, stress level information. However the signal amplitude decreases for both tensile and compressive stresses making evaluation ambiguous (Fig. 3). By using MAE together with SMA, absolute biaxial measurement can be achieved. A unique development of the technique has been to measure variations of stress up to a depth of 10 mm. 6

2.4 Magnetically Induced Velocity Change (MIVC) Ultrasonic velocity has a small sensitivity to the magnetic field strength in the material as well as the internal stress state. Both bulk and surface stresses can be investigated by using either bulk longitudinal and shear waves or surface ultrasonic waves. The MIVC values are less sensitive to preferred grain alignment than ultrasonic stress measurement methods because the technique is based upon the magnetoelastic effect rather than acoustoelasticity. 2.5 Magnetic Technique Combinations In order to improve accuracy of measurement and reduce the influence of factors such as microstructure, researchers have used combinations of two or more magnetic methods. These are usually incorporated into a single instrument to ease use and maintain a rapid method. One example of a technique combination (MAE and SMA) has already been referred to above. Some workers have combined BE with MAE, the authors have combined SMA with DEP, while one commercially available system combines BE, incremental permeability, coercivity, and NLHA (Schneider 1998). Methods involving more than two techniques often use empirical rather than analytical means of data interpretation.

Residual Stresses: Measurement using Magnetoelastic Effects 3. Materials Issues The magnetic properties of materials are affected by the material composition and structure. It is important to understand these effects when using magnetic methods to measure stress, since to ignore them is to risk misinterpreting results. At a fundamental level the alloy content will affect whether the material is ferromagnetic, and the magnitude of the permeability. Thus alloys which are close to 100% iron or nickel with unpaired d-electrons are strongly ferromagnetic, but alloys with intermediate concentrations of iron or nickel with other elements may be either ferromagnetic or not, depending on the filling of the d-band. The alloy composition also determines whether the structure is f.c.c. or b.c.c., which in turn affects the magnetic properties. In steels, the carbon content is important mainly for determining magnetic softness or hardness because the carbon tends to cause domain wall pinning whether in solution or precipitate form. Thus the sensitivity of a magnetic parameter to stress varies with alloy composition. The crystal structure can have an important effect on magnetic properties via crystallographic texture. If a grain of the material has a high crystal anisotropy then it will also have a high magnetic anisotropy. If stress measurement is attempted in a region of strongly preferred orientation, for example in a rolled material that has not fully recrystallized, then the texture must be compensated for to avoid errors. Domain walls are pinned by various microstructural features introduced during fabrication, including grain boundaries, phase boundaries, precipitates, inclusions, and dislocations. In the first case if the grain size decreases there will be more pinning points for the domain walls. In the second case, it will depend upon the nature of the phases present. If only one of the phases is magnetic, or if there is a large difference in properties, then the phase boundary will strongly pin the walls, raise the coercivity, and cause the hysteresis loop to broaden and the material to harden magnetically, e.g., in pearlite with alternating ferrite and cementite (iron carbide) laths. In the case of precipitates or inclusions, it depends on their magnetic properties. If they are nonmagnetic, as is often the case in steels, then they pin the domain walls strongly, raising the coercivity and hardening the structure magnetically (Buttle et al. 1987b). Thus the sensitivity of a particular magnetic measurement to stress can change during heat treatments such as annealing and aging, and may vary from specimen to specimen of nominally the same material. An important example arises near welds where the heat affected zone (HAZ) will contain significant variations in microstructure just where knowledge of residual stress levels may be desirable. Figure 4 shows how the stress sensitivity of two magnetic methods is modified as the microstructure changes from ferrite\pearlite to bainite and finally martensite. Care must be taken to ensure that

material effects such as these are compensated for when using that measurement to deduce the stress state. Thermomechanical treatment and service loading may mean the material has been plastically deformed, in which case it will contain networks of dislocations. These too may pin the magnetic domain walls, although the effect is not great unless there has been heavy deformation or fatigue (Maker and Tanner 1998). In practice it is not always possible to fully separate microstructural from stress effects. One example is during the case hardening of steel to produce a martensitic surface layer. Martensite has a fine lath structure with dissolved carbon. This structure strongly pins domain walls. However, because of its different crystal structure, martensite is heavily strained, so that there is an internally generated residual stress that will change the local permeability. It is impossible to separate these two effects with a simple measurement. A second example is the bending of a bar. This generates an intrinsic depth dependent combination of plastic and elastic strains, which will remain unless the bar is annealed. A simple magnetic measurement will be sensitive to both simultaneously.

4. Practical Issues The microstructural sensitivity of the magnetic parameters makes it important to carry out calibration. For some methods, such as those based on anisotropy, one calibration will serve over a significant range of material conditions. Other methods may be highly sensitive to variations in microstructure, so that it may not be possible to obtain a unique calibration for an application. Calibration is usually carried out by applying known stresses to test samples. However, it is important to ensure that the initial stress state of the test samples is known. Since magnetic techniques exhibit a tensor relationship to stress, a uniaxial test is usually not sufficient to characterize the material response. The desired spatial resolution, measurement penetration, measurement time, and accuracy must all be considered when designing probes. The requirements are interrelated so that it is necessary to derive a compromise between them. Small probes having high spatial resolution, obviously cannot measure deep below the surface but they are also much more sensitive to small variations in lift-off. For some of the most important applications on components and pipework, it is necessary to measure on curved surfaces. Here some of the issues in probe design are more acute. Smaller probes may be necessary for restricted space, or alternatively shaping the probes for the geometry may be possible. However, shaped probes cannot be rotated to determine the 7

Residual Stresses: Measurement using Magnetoelastic Effects stress axes. Electromagnetic finite element software has been used to design probes in some cases. For techniques that use applied fields well below saturation, remanent fields are likely to influence the measurements. In this case it may be necessary to use a procedure to demagnetize the material or at least remove the magnetic history by randomizing the domains at the measurement positions.

ever, magnetic methods are beginning to be better understood in terms of their relationship to stress, and new multi-parameter systems are becoming available. Their speed of operation, controlled depth penetration, and growing track record point the way to greater industrial use in the future.

Bibliography 5. Examples of Residual Stress Measurement Magnetic stress measurement has been used for a wide range of different industrial applications from fabrication through to in-service plant integrity. A simple laboratory example is the plastically bent bar (Fig. 5). The positron annihilation technique was used to confirm significant levels of plastic damage close to the top and bottom surfaces of the bar. SMA measurements indicated residual elastic strains (residual stresses), which followed the characteristic (j,k,j,k) variation across the side face. Figure 6 shows an example from a fabrication process using the MAE and SMA methods on a 25 mm butt welded mild steel plate. Here comparison is made with the semi-destructive center-hole method. In another fabrication example the technique combination, SMA and DEP, was applied to heavily eroded forged turbine blades taken from a low pressure steam turbine. In addition, two examples showing the measurement of depth dependent stress is shown in Fig. 7.

6. Conclusions There are many magnetic methods of stress measurement, each having a different response to stress and material microstructure. Their complexity has tended to limit successful application in the past, despite the rapid advances in nondestructive technology. How-

Allen A J, Buttle D J, Dalzell W, Hutchings M T 2000 Residual stress in butt weldments of 50D steel measured by neutron diffraction and magnetic techniques. In: Proc. Int. Conf. Residual Stress. IOM Communications, p. 6 Buttle D J, Scruby C B, Briggs G A D, Jakubovics J P 1987a The measurement of stress in steels of varying microstructure by magnetoacoustic and Barkhausen emission. Proc. R. Soc. London Ser. A 414, 469–97 Buttle D J, Scruby C B, Jakubovics J P, Briggs G A D 1987b Magneto-acoustic and Barkhausen emission from domain wall interactions with precipitates in Incoloy 904. Phil. Mag. A 55, 735–56 Hauser H 1994 Energetic model of ferromagnetic hysteresis. J. Appl. Phys. 75, 2584–97 Jiles D C 1988 Review of magnetic methods for nondestructive evaluation. NDT Int. 21, 311–9 Kwan H 1986 A nondestructive measurement of residual bulk stresses in welded steel specimens by use of magnetically induced velocity changes for ultrasonic waves. Mater. EŠal. 44, 1560–6 Maker J M, Tanner B K 1998 The in situ measurement of the effect of plastic deformation on the magnetic properties of steel. Part I. Hysteresis loops and magnetization. J. Magn. Magn. Mater. 184, 193–208 Sablik M J, Riley L A, Burkhardt G L, Kwan H, Cannell P Y, Watts K T, Langman R A 1994 Micromagnetic model for biaxial stress effects on magnetic properties. J. Magn. Magn. Mater. 132, 131–48 Schneider E 1998. Nondestructive analysis of stress states in components using micromagnetic and ultrasonic techniques—an overview. Proc. ECNDT 98, 3, No.11, Copenhagen, Denmark, 26–29 May 1998

D. Buttle and C. Scruby

Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 8172–8180 8