Multiproperty Microstructure and Property Design of ... - CiteSeerX

Dongsheng Li School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive, Atlanta, GA 30332

Anthony D. Rollett Department of Materials Science and Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213

Greg Vialle Hamid Garmestani School of Materials Science and Engineering, Georgia Institute of Technology, 771 Ferst Drive, Atlanta, GA 30332

1

Multiproperty Microstructure and Property Design of Magnetic Materials Microstructure sensitive design was used in this study to design a textured soft magnet material to meet a range of magnetic properties. The evolution of microstructure and magnetic properties during mechanical processing was simulated and presented in a spectral representation for microstructure (texture hull) and magnetic property (property hull). The set of properties for a single path (or multiple processing paths) is represented in the property hull with a direct link to the range of desired microstructures. A methodology is proposed to achieve microstructures satisfying the requirement of multiple properties. 关DOI: 10.1115/1.2870235兴 Keywords: materials design, processing path, property hull, magnetic permeability, Green’s function

Introduction

The present modeling tools rely heavily on prior experimental results and conventional metallurgical methodologies based on phase diagrams and alloy additions 关1–6兴. Such approaches require a large number of experiments and in many cases, the results are specific to the materials composition and thermomechanical or chemical process. The aim of the analytical tools developed in materials science has been to reduce the number of the experiments and provide the designer or the materials engineer with a design tool that can be used to identify newer processes and alternative routes. To design a material 共and its microstructure兲 is to try to meet a material user’s needs by tailoring its chemical and internal structure 关7兴. Materials design is inverted from the age-old approach to materials, which is described as deductive, cause-effect in its logic, and proceeding from processing to performance. What is sought is an approach that flows from the goals of the design team 共properties, performance兲, through specification of the chemistry and microstructure of the candidate material, and, finally, to a description of the microstructure and the processing path. While our methods can be applied to a wide range of materials with different applications 关4,6,8–14兴, we will concentrate our efforts on the thermomechanical treatment. Material design charts have been used extensively by designers as an aid to material selection for many years. Data regarding material properties have been compiled into Ashby-type charts. The attempt to link property-microstructure processing into a unified framework as a basis for a new paradigm in materials science, such as Fig. 1, has now been recognized as a new paradigm in materials science known as microstructure sensitive design 共MSD兲关8,15兴. This framework allows for ready selection of optimal material/property combinations. MSD methodology in its original form uses a spectral representation as a tool to allow the mechanical design to take advantage of the microstructure as a continuous design variable 关8,15兴. It uses a set of Fourier basis Contributed by the Materials Division of ASME for publication in the JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received August 16, 2007; final manuscript received January 17, 2008; published online April 2, 2008. Assoc. Editor: Matthew P. Miller. Paper presented at the Fifth International Conference on Materials Processing Defects 共MPD-5兲.

functions to represent the microstructure 共e.g., single orientations兲 as the material set 关8,15兴. The combination of all these elements of microstructure states can be used to construct the property enclosure for any particular structure. The procedure in this methodology can be summarized as follows: 共a兲

Microstructure representation: The microstructure and its details are represented by a set of orthogonal basis functions ␹n. F共␹n,Cn兲 =

兺C ␹

n n

共1兲

n

共b兲

where Cn’s are the coefficients, determined for each individual microstructure. Properties and constraints: The properties and constraints are represented in the same orthogonal space P共␹n,pn兲 =

兺p ␹

n n

共2兲

n

共c兲

共d兲

Coupling: The properties and constraints can be represented by hyperplanes in the property enclosure, which is defined as a universe of all variations in the relationships among several properties for the same microstructure. Designer materials: Intersection of these planes defines the universe of all materials and microstructure 共distributions兲 appropriate for design. This is similar to how Ashby’s diagrams are being used in design 关16兴.

In the present paper, a similar methodology is developed and applied to soft magnet polycrystalline materials for multiproperty design. MSD is presently taking advantage of “texture” in the form of orientation distribution function for the representation of polycrystalline materials 关8,15兴. The orientation distribution function 共ODF兲 is a one-point statistical distribution function that only considers volume fractions 共or number fractions兲 of crystallites with the same orientation. Two-point statistical functions can be used as a first order correction to the average representation 关3–6兴. Two-point correlation functions provide information about near neighbor and far field effects and allow the defect sensitive properties to be incorporated in the analysis. The extension to higher order statistics adds a higher order of dimensionality in the mate-

Journal of Engineering Materials and Technology Copyright © 2008 by ASME

APRIL 2008, Vol. 130 / 021023-1

Downloaded 04 Jul 2008 to 194.57.144.98. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Fig. 1 Materials design paradigm, showing the inverse flow from the design to property requirement to microstructure and to the last goal, processing

rials hull. It also presents two major improvements in the analysis for the calculation of effective properties and the evolution of microstructures. The composite formulation will be markedly enhanced by the use of two-point correlations. Here, we take advantage of anisotropy in the microstructure of a polycrystalline Fe material as a parameter for design. The methodology presented here is an application of a framework for “MSD” for the case of composites. The earlier work in this effort concentrated on polycrystalline materials. The main issue within this methodology is the limited microstructure representation used in the earlier work 关8,15兴. A one-point statistical function 共ODF兲 was used in the form of spherical harmonics. In the present paper, a two-point statistical function representation is used for the representation of the microstructure of a two-phase composite. Such a representation will allow us to link the microstructure to mechanical 共elastic and plastic兲 and magnetic properties.

2

Microstructure Representation

再

x 苸 Vi

1

0 otherwise

冎

共3a兲

By definition, the volume fraction for phase i, vi, is considered to be a one-point correlation function

␾i = P兵L 共x兲 = 1其 i

共3b兲

It is clear that volume fraction alone cannot capture the whole complexity of morphology in a random heterogeneous medium when studying effective properties. One example is the difference observed when two bounding theories, series and parallel, are used, respectively, in the prediction of properties for a composite with the same volume fraction. More details of the shape and morphology of the microstructure including the interaction of the components in the media and orientation distribution of crystallographic grains 共texture兲 should be considered in order to give a reasonable prediction of effective properties. This can only be realized by using higher order distribution functions 关2,17,18兴. A two-point distribution function is defined as a conditional probability function when the statistics of a three-dimensional vector, r = r2 − r1, is investigated once attached to each set of random points in a particular microstructure: Pi1i2共r兲 = P兵L 共r1兲 = 1,L 共r2兲 = 1其 i1

i2

共4兲

Here, Pi1i2共r2 − r1兲 is the probability of the event r with vector r1 in phase i1 and vector r2 in phase i2. It should be noticed that in many cases, the medium is anisotropic and it is inappropriate to simplify the vector to a scalar parameter. 021023-2 / Vol. 130, APRIL 2008

The most popularly used formula to represent the distribution function is an exponential function proposed by Corson 关17兴. Pij共r兲 =

In this framework, the statistical details of a heterogeneous medium can be represented by an n-point probability distribution function 关3–6兴. Statistical descriptions can give more information about the microstructure, such as disorientation, grain size, and so on. A polycrystalline material may be considered as a material that is composed of different components, each with a specified range of orientation. Let us divide Euler space into N3 subspaces. An indicator function Li共x兲 for component i is used to identify a random point x, located inside or outside of component i: Li共x兲 =

Fig. 2 Correlation function parameter cij at different angles ␪ for a porous YSZ microstructure shown in the embedded micrograph. Fit curve gives parameters cij0 and A in modified Corson’s function.

再

viv j − viv j exp共− cijrnij兲 vi2

i⫽j

+ vi共1 − vi兲exp共− cijrnij兲 i = j

冎

共5兲

For a two-phase composite 共including porous materials兲, i and j correspond to Phases 1 and 2. The constants cij and nij are microstructure parameters obtained by fitting the real correlation curve. This relationship has been shown to be appropriate for random isotropic microstructures. For anisotropic case, the direction of vector r should be taken into consideration. Saheli et al. 关14兴 introduced a simplified form of an anisotropic two-point correlation function with nij as 1 and used it for an axially symmetric sample. In that study, the empirical coefficient cij, a scaling parameter representing the correlation distance, is reformulated by a Fourier expansion. Taking into account only the first order term, cij共␪,K兲 = cij0 共1 + 共1 − K兲sin ␪兲

共6兲

where K is a material parameter that represents the degree of anisotropy in a microstructure such that K = 1 corresponds to an isotropic microstructure, and c0ij is the reference empirical coefficient. This formulation was used to characterize microstructure of a porous material composed of yttrium stabilized zirconia 共YSZ兲 as Phase 1 and voids as Phase 2. The evolution of parameter cij with direction 共represented by sin ␪兲 is shown in Fig. 2. A linear fit of these points gives two correlation parameters for the microstructure shown in the embedded micrograph: c0ij = 0.67 and K = 0.45. K is an indicator of anisotropy in the range of 0 and 1. The smaller K represents higher anisotropy. In this study, for an anisotropic heterogeneous sample, a three dimensional form of the correlation function is proposed. Pij共r共r, ␪, ␾兲兲 =

再

viv j − viv j exp共− cij共␪, ␾兲r兲

i⫽j

vi2 + vi共1 − vi兲exp共− cij共␪, ␾兲r兲 i = j

冎

共7兲

The length of vector r is given by a scalar r and its direction given by ␪ and ␾ in a spherical sample coordinate system. We will use this formula in the next section to represent the microstructure of heterogeneous media. Transactions of the ASME


3

Simulation of Magnetic Property

␸共x兲 = ␸0共x兲 −

We assume that heterogeneous media are composed of n constituents with different permeability ␮i 共i = 1 , . . . , n兲 and partitions vi. The local magnetic flux density B and local magnetic field H at any arbitrary point x will satisfy the linear relationship such that Bi共x兲 = ␮ij共x兲Hk共x兲

共8兲

This relationship may only be valid during the initial magnetization and for most practical purposes, the nonlinear relation should be considered. In the case of inductor materials and at high frequency domains, the field generated is extremely low and a linear relation may be valid. When the polycrystalline alloy is placed in a small magnetic field such as 10 Oe, the field is not strong enough to rotate the local magnetic moment out of the easy direction. The magnetization field will distribute along the six 具100典 easy magnetization axes to avoid high demagnetization energy. The net component of magnetization in a single crystal Bi along applied field is a function of the orientation 关19–21兴 1 Bi = Bs ␣1 + ␣2 + ␣3

g共x,x⬘兲 =

˜ 共x⬘兲 dx⬘ ⵜ g共x,x⬘兲P

共19兲

1 1 4␲␮0 x − x⬘

共20兲

where f 0共x兲 is the potential field at infinity. To obtain the field H, Eq. 共17兲 is differentiated: H共x兲 = H0 +

冕

˜ 共x⬘兲 dx⬘G共x − x⬘兲P

H共x兲 = H0 +

冕

˜ 共H0,h共x⬘兲兲H0 dx⬘G共x − x⬘兲␮

具H共x兲典h = H0 +

冕

˜ 共H0,h共x⬘兲兲典hH0 dx⬘G共x − x⬘兲具␮

˜ 共E0,h共x⬘兲兲典h = 具␮

冕

˜ 共H0,h共x⬘兲兲dh共r⬘兲 f共r⬘ 苸 h共r⬘兲兩r 苸 h兲␮ 共24兲

where the conditional two-point correlation function f共r⬘ 苸 h共r⬘兲兩 r 苸 h兲 is defined as the probability of occurrence of r⬘ at state h共r⬘兲 given that r belongs to state h:

␣2 = 兩共− cos ␾1 sin ␾2 − sin ␾1 cos ␾2 cos ␾兲cos ␥ + 共− sin ␾1 sin ␾2 + cos ␾1 cos ␾2 cos ␾兲sin ␥兩共10兲

The effective permeability ␮eff in the heterogeneous medium is defined by 共11兲

where the symbol 具¯典 denotes the ensemble average. To define the relationship between the localized permeability ␮共x兲 and the ensemble average of the permeability ␮0, we introduce polarized ˜ 共x兲 such that permeability ␮ ˜ 共x兲 ␮共x兲 = ␮0 + ␮

共12兲

˜ 共x兲H共x兲 B共x兲 = ␮0H共x兲 + ␮

共13兲

then we have If we define the polarized field ˜P共x兲 as ˜P共x兲 = ␮ ˜ 共x兲H共x兲

共14兲

B共x兲 = ␮0H共x兲 + ˜P共x兲

共15兲

then we have

Since the magnetic flux is divergence free, ⵜ · B共x兲 = 0 = ␮0 ⵜ · H共x兲 + ⵜ · ˜P共x兲

共16兲

Define a potential field ␸ such that H = − ⵜ␸

共17兲共18兲

Equation 共18兲 is a set of partial differential equations that can be solved using a number of techniques. Using Green’s function 关18兴, the solution of Eq. 共18兲 is given as Journal of Engineering Materials and Technology

f共r⬘ 苸 h j兩r 苸 hi兲 = Pij/Vi

共25兲

After the statistical localized field, 具H共x兲典h, is obtained, the corresponding statistical localized current density, 具B共x兲典h, is calculated: 具B共x兲典h = ␮h具H共x兲典h

共26兲

Since no assumption is used on representing the statistical distribution of components in the heterogeneous medium, the application of this statistical continuum model to the prediction of thermal conductivity can cover a broad range of materials systems.

4

Processing Path Model

Processing path model, developed by Li et al. 关12,13兴, is based on a conservative principle proposed by Clement and co-worker 关22,23兴. Microstructure, represented by texture f共g兲, the crystal orientation distribution, is described as a function of a processing metric ␩. Texture can be expressed as a series of generalized spherical harmonic functions in which each Fmn is the weight l 共coefficient兲 associated with a corresponding harmonic, as shown in Eq. 共27兲: ⬁ M 共l兲 N共l兲

f共g, ␩兲 =

兺兺兺F

mn ¨˙ mn l 共␩兲Tl 共g兲

共27兲

l=0 m=0 n=0

Substitute Eq. 共17兲 back to Eq. 共16兲,

␮0 ⵜ · 共ⵜ␸兲 = ⵜ · ˜P共x兲

共23兲

˜ 共H0 , h共x 兲兲典h can be described in The correlation function 具␮ ⬘ terms of the conditional two-point probability density function of state h,

+ cos ␾1 sin ␾2 cos ␾兲sin ␥兩

具B共x兲典 = ␮eff具H共x兲典

共22兲

The average field for state h can be calculated from the above equation:

␣1 = 兩共cos ␾1 cos ␾2 − sin ␾1 sin ␾2 cos ␾兲cos ␥ + 共sin ␾1 cos ␾2

␣3 = 兩sin ␾1 sin ␾ cos ␥ + − cos ␾1 sin ␾ sin ␥兩

共21兲

where H0 is the applied magnetic field. Rewriting the above equation, we get

共9兲

Here, Bi is the magnetization of the ith single crystal and Bs is the saturation magnetization. The three parameters, ␣1, ␣2, ␣3, are direction cosines of the easy directions with applied magnetic field. They may be obtained from the angle of the applied magnetic field 共say, ␥ with the rolling direction兲 and the orientation of the crystallite g represented by three Euler angles 兵␾1 , ␾ , ␾2其:

冕

˙ T¨mn l 共g兲 are the symmetric generalized spherical harmonics for the corresponding sample and crystal symmetry. For a single crystal orientation distribution in which all the crystals are oriented along gi, texture coefficients Fmn l are calculated directly from the spherical harmonics: ˙ Flmn = 共2l + 1兲T¨lmn共gi兲

共28兲

APRIL 2008, Vol. 130 / 021023-3


In Clement’s work 关22,23兴, the texture evolution is regarded as a fluid flow in orientation space. Three Eulerian angles compose the orthogonal coordinates of this space. For any point represented by g, the density is 共1 / 8␲2兲f共g , ␩兲sin ␾ and the flow rate is R共g兲. According to the conservation principle in the orientation space, the sum of the increase of the quantity of matter in an element of volume dv and the material moving across the surface S should be zero. The continuity equation can then be represented as

冖冉冊 f共g, ␩兲

S

1 ⳵ ˆ 2 sin ␾R · nd␴ + 8␲ ⳵t

冕冕冕冉冊 f共g, ␩兲

V

1 sin ␾dv 8␲2 共29兲

=0 Thus,

⳵ f共g, ␩兲 1 div关f共g, ␩兲sin ␾R共g兲兴 = 0 + ⳵t sin ␾

共30兲

For an infinitesimal volume element in the orientation space, dv = d␸1d␸d␸2, the first term in Eq. 共30兲 is the increase of the quantity of matter per unit time and the second term describes the quantity of matter moving out of the infinitesimal volume element. By simplifying Eq. 共30兲, the continuity equation for the conservation of quantity of matter in a volume element in the Euler space is

⳵ f共g, ␩兲 + div关f共g, ␩兲R共g兲兴 + cot ␾ f共g, ␩兲R共g兲 = 0 ⳵t

共31兲

Using the expression for texture in Eq. 共27兲, Eq. 共31兲 is expanded in a series of spherical harmonics:

兺 lmn

dFlmn共␩兲 ¨˙ mn ˙ Tl 共g兲 + F␭␴␳共␩兲共div共T¨␭␴␳共g兲R共g兲兲 d␩ ␭␴␳

兺

˙ + cot ␾T¨␭␴␳共g兲R共g兲兲 = 0

共32兲

The second summation can be further expanded into a series of generalized spherical harmonics: ˙ ˙ div共T¨␭␴␳共g兲R共g兲兲 + cot ␾T¨␭␴␳共g兲R共g兲 = −

兺A

mn␴␳ ¨˙ mn Tl 共g兲 l␭

lmn

共33兲 ␴␳mn A␭l

is introduced as the coefficients of the spherical Here, harmonics. Substituting this back into Eq. 共32兲, a linear relationship between the texture coefficients and their rate of change is derived: dFlmn共␩兲 mn␴␳ ␴␳ = Al␭ F␭ 共␩兲 d␩ ␭␴␳

兺

共34兲

This linear relationship was used by Bunge and Esling 关24兴 and Klein and Bunge 关25兴 to predict texture evolution in orientation space. In this work, a texture evolution function obtained by the integration of Eq. 共34兲 was used to describe the evolution of the texture coefficients with the deformation parameter: F共␩兲 = eA␩F共␩0 = 0兲

共35兲

The coefficients represented by the sixth rank tensor A can be rearranged as the elements of a matrix and will be called “texture evolution matrix” in this work. The introduction of a microstructure parameter A can be used as a methodology to predict microstructure evolution in a large set of polycrystalline materials. The sixth order tensor A, as derived in Eq. 共34兲共continuity relation in the Euler space兲, is the underlying parameter in this basic law. The sixth order tensor may be a function of many other different features of the microstructure. It is clear that the evolution of texture is a function of the details and physics of the microstructure and the underlying deformation mechanism. At a first glance, it seems that such details 021023-4 / Vol. 130, APRIL 2008

are neglected in the formulation presented in the paper. All these details are, however, embedded in the sixth order tensor. Texture evolution matrix is an intrinsic property to a monotonic processing method and specific deformation mechanism. If the deformation mechanism changes during the deformation, for example, other slip systems requiring higher activation energy are activated at large strain, the texture evolution matrix also changes. To obtain the texture evolution matrix A, we may use experimental data or simulated results. Since a large number of input data are needed, simulated results based on an existing successful model will be used. In this work, a modified Taylor model developed by Kalidindi and Anand 关26兴 was used to calculate A and further processing path. 24 slip systems in the 具111典兵110其 and 具111典兵112其 groups were assumed to be active in the deformation of bcc iron.

5 Application of Process Design Under MultipleProperty Requirement A randomly textured iron-based material is chosen for the purposes of this work. The processing tools described above allow us to process this material in a variety of loading conditions to obtain the final microstructure of interest for the desired soft magnetic properties. As is clear in the above section, there is a strong linkage between magnetic properties and texture. The properties are then simulated using the homogenization relations based on the two-point function statistics. On the other hand, the processing path model simulates texture evolution during mechanical processing. The application of texture evolution path in MSD is more understandable when constructed in the texture hull. Figur. 3共a兲 shows that the materials 共texture兲 hull is a compact convex subspace. All possible textures 共representing microstructures兲 reside within this convex subspace such that points outside the wireframed texture hull are unphysical. Microstructure is represented by a point in the microstructure hull composed of the first three nonzero coefficients F411, F412, and F413. This representation does not ignore the existence of the other coefficients and can be considered as a three dimensional projection of an n-dimensional space. Texture coefficients are characterized by three orders, l, m, and n, as shown in Eq. 共27兲. To correlate the microstructure and properties 共second order tensor兲, texture coefficients with order l not larger than 4 are sufficient. For cubic crystal system 共iron兲 and orthotropic sample symmetry, there are only four nonzero texture coefficients when the order l is not bigger than 4. Since one of the four coefficients, F011, is a constant and equal to 1, three texture coefficients, F411, F412, and F413, are enough to represent the microstructure when second order tensor properties are considered. Figure 3共b兲 shows how microstructure in the initial sample evolves in the microstructure hull during three different deformation modes. It should be noted that the scales used in Figs. 3共a兲 and 3共b兲 are different. The evolutions of texture during different deformation modes shown in Fig. 3共b兲 only occupy a very small portion of microstructure hull in Fig. 3共a兲. A small black line close to the origin in Fig. 3共a兲 represents the texture evolution during rolling, which is magnified in Fig. 3共b兲. In Fig. 3共b兲, the red line represents the processing path under rolling along the transverse direction to a strain of 100%. The blue line is for uniaxial tension along the axial direction to a strain of 100% and the green line shows uniaxial compression along the axial direction to the same strain of 100%. They all start from the same point representing the microstructure of the initial sample in the texture hull. Compared with Figs. 3共a兲 and 3共b兲, it is evident that the microstructure evolution only occupies a small portion of the microstructure hull, even under a large strain of 100% in all these three deformation modes. This fact demonstrates how little set of microstructure we have studied for polycrystalline materials and how large space of properties has been left for us to exploit. To achieve other microstructures, a combination of deformation modes may be considered. Another solution is to use thermal processing 共annealing兲 to change texture by annealing and recrystallization. Transactions of the ASME


(a)

Fig. 4 Property hull composed of permeability tensor components ␮11 and ␮22. All the dot points correspond to the property of single crystals at different orientations with Euler angles at intermediate of 5 deg: „f1 = 0 , 5 , . . . , 90; f = 0 , 5 , . . . , 90; f2 = 0 , 5 , . . . , 90…. Critical points on the boundary of the property hull are point A at „2100, 2100… for orientation „001…Š100‹, point B at „1485, 2100… for orientation „011…Š01¯ 1‹, point C at „1240, 1240… for orientation „111…Š11¯ 0‹, and point D at „2100, 1485… for orientation „101…Š101¯ ‹.

(b) Fig. 3 „a… Microstructure hull shown as a wired frame in the materials space composed by three texture coefficients; „b… processing paths of microstructure evolution in the materials space composed by F411, F412, and F413 during different processing methods, rolling, compression, and tension

The property hull for cubic iron structures is investigated in this study to discover the range of magnetic properties that can be achieved by adjusting the microstructure without a change in composition. The property hull was obtained by investigating all the possible magnetic properties of the single crystal orientations 共and their combinations兲 in a polycrystalline 共textured兲 sample. Figure 4 gives permeability tensor components, ␮11 and ␮22, for single crystals at different orientations with Euler angles at intermediate steps of 5 deg: 共␾1 = 0 , 5 , . . . , 90; ␾ = 0 , 5 , . . . , 90; ␾2 = 0 , 5 , . . . , 90兲. Some critical points on the boundary of the property hull are identified below. The upper right point, A, is 共2100, 2100兲 for orientation 共001兲具100典. The upper left point, B, is 共1485, ¯ 1典. The corresponding lower right 2100兲 for orientation 共011兲具01 ¯ 典. The lower point, D, is 共2100, 1485兲 for orientation 共101兲具101 ¯ 0典. The left point, C, is at 共1240, 1240兲 for orientation 共111兲具11 upper point, A, corresponds to the easy axis of magnetization and the lower point, C, corresponds to the hard axis of magnetization. Journal of Engineering Materials and Technology

The second order tensor magnetic permeability was simulated from the microstructure based on the framework of statistical mechanics. Evolution of the diagonal elements of permeability tensor, ␮11, ␮22, and ␮33 for the random iron choke, during rolling, uniaxial compression, and tension is illustrated in Figs. 5共a兲–5共c兲, respectively. ␮11, ␮22, and ␮33 characterize the permeability along the hoop direction, radial direction, and axial direction, respectively. It is shown that the property changes significantly at the initial small strain and quickly saturates after a strain of 20%, although the microstructure continues to evolve significantly, as shown in Fig. 3共b兲. The goal of the materials design is to discover a processing path to achieve the required property requirements. Figure 6 illustrates the property evolution up to a strain of 100% during rolling, compression, and tension, respectively. Considering the machinability and the saturation of texture evolution during a single deformation mode, property evolution at strains larger than 100% is not presented. The properties studied are permeability tensor components ␮11 and ␮22. Just as the evolution of microstructure is confined in the microstructure hull shown in Fig. 3共a兲, the evolution of these properties is also confined in a convex subspace, the property hull. The properties achieved even at a large strain for these three deformation modes only represent a small portion of the property hull, as shown in Fig. 5共a兲. This corresponds to Fig. 3共b兲, which demonstrates that only a small fraction of the possible microstructures will be achieved by these three deformation modes. Single mode of deformation demonstrated here will not reach a large range of microstructure and, as a consequence, will not give a large range of properties. Combination of deformation modes or other processing methods may reach a large part of microstructure and corresponding properties. The property evolution curves shown in Fig. 6 are not straight lines. This trend is also illustrated in the processing paths shown in Fig. 3共b兲. Figure 6 also demonstrates how this processing path will help materials designer in deciding which property ranges may be obtained and how to achieve them. Figure 6共b兲 is a magnified part of Fig. 6共a兲. The APRIL 2008, Vol. 130 / 021023-5


(a)

(a)

(b)

(b)

(c) Fig. 5 Evolution of magnetic permeability tensor elements during „a… rolling, „b… uniaxial compression, and „c… uniaxial tension

initial status, Point A, is a random texture. The blue dot curve represents the property evolution during uniaxial tension. The black solid curve represents evolution during rolling, and the red dot curve represents compression. Corresponding property points at strains of 50% and 100% are labeled. If the design requirement for permeabilities ␮11 and ␮22 falls into the red circle, for example, ␮11 艌 1550艚 ␮22 艌 1550, then it is achievable by rolling or uniaxial compression to a strain of 100%. Uniaxial tension will not satisfy the requirements of magnetic permeability. In another case, if the design requirement for permeabilities ␮11 and ␮22 falls into the blue circle, for example, ␮22 艌 1580, then the desired microstructure can be produced only by rolling to a strain of 50%.

6

Conclusions

MSD has been applied to a polycrystalline iron material system. The key to MSD is the correct representation of the microstructure. The statistical formulation uses the two-point statistical functions to incorporate the effect of the microstructure distribution. In the case of anisotropic distribution, the two-point function can introduce anisotropy in the effective magnetic permeability. Such anisotropy can be used as a parameter to engineer new composites with an imposed distribution. These parameters and their concomitant properties are considered to be continuous design variables that can be used for process design of an iron-based choke. 021023-6 / Vol. 130, APRIL 2008

Fig. 6 „a… Evolution of magnetic properties during processing in property space. „b… Magnified processing paths for random samples under rolling, compression, and tension, respectively. Point A is the starting point for random texture sample. Points B, C, and D are property points for samples at a strain of 100% under tension, rolling, and compression, respectively. If the design goal falls in the blue circle, rolling will achieve that property requirement. If the design goal falls in the red circle, rolling and compression both will satisfy.

An example is provided such that the design constraint can be reduced to a set of microstructures. The design objectives and constraints are then represented in a specific property space that accounts for multiple properties.

Acknowledgments This work has been funded under AFOSR Grant No. F4962003-1-0011 and Army Research Lab Contract Nos. DAAD17-02P-0398 and DAAD 19-01-1-0742.

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APRIL 2008, Vol. 130 / 021023-7


Multiproperty Microstructure and Property Design of ... - CiteSeerX

Multiproperty Microstructure and Property Design of ... - CiteSeerX

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