Exact connections between effective magnetostriction and effective elastic moduli of fibrous composites and polycrystals

Exact connections between effective magnetostriction and effective elastic moduli of fibrous composites and polycrystals

T. Chen

^a)

Department of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan, Republic of China

C.-W. Nan

Department of Materials Science & Engineering, Tsinghua University, Beijing 100084, People’s Republic of China

G. J. Weng

Department of Mechanical & Aerospace Engineering, Rutgers University, New Brunswick, New Jersey 08903

共Received 2 January 2003; accepted 27 March 2003兲

The effective magnetostriction of a two-phase fibrous composite and a two-dimensional polycrystal assembled from cylindrical magnetostrictive cubic crystals is studied. For the considered systems, we show that there exist exact microstructure-independent relations between the effective magnetostriction and the effective elastic moduli. The fibers could be aligned identically or randomly oriented in the transverse plane. There is no restriction on the cross-sectional shapes of the fibers, nor on the arrangement of transverse geometry of the composite aggregate. These connections imply that knowledge of the effective elastic moduli will readily provide the effective magnetostriction of the composite medium. © 2003 American Institute of Physics.

关DOI: 10.1063/1.1576512兴

I. INTRODUCTION

When an object is subjected to a magnetic field, its di- mensions change. This deformation is referred to as a mag- netostriction strain, denoted by

␭.¹

The magnetostriction

␭

varies nonlinearly with the applied magnetic field H and at- tains a saturation value for a sufficiently large H.

²

In this work, we are concerned with the magnetostrictive response associated with the saturation state of composite aggregates.

Magnetostrictive composites are mixtures of a magnetostric- tive phase together with a nonmagnetostrictive

共or magneto-

strictive

兲 matrix to gain magnetostrictive effect and, at the

same time, to possess good mechanical toughness and strength. These composites are of potential applications in a few technological devices, such as magnetoactive sensors and actuators. To gain insights into the understanding of their physical behavior, it is important to characterize the overall magnetostriction of composites in terms of their constituent properties and microstructure information in a rigorous man- ner. In the analysis, typically one may regard the magneto- striction as a stress free strain due to a magnetic field, much like the spirit of the thermal strain induced from the uniform temperature change. Nevertheless, the determination of the effective magnetostriction and the effective thermal expan- sion of composites is not exactly alike. The magnetostriction is intrinsically orientation dependent, while the thermal strain is not. Much progress has been made in the last few years, including some experimental works.

^3–5

Among the analytical studies, one branch is directed toward the deriva-

tion of the upper and lower bounds for the effective magne- tostriction of polycrystals

⁵

and composites. The other branch is to provide an estimate for the effective

␭ based on various

micromechanical schemes.

^{6 –10}

This article, on the other hand, is to pursue exact microstructure independent relations on the effective magnetostriction. Specifically, we find that there are situations in which the effective magnetostriction can be exactly linked with the effective elastic moduli of a composite medium. Explicit connections are derived for a two-phase composite made of an elastic matrix with fibrous magnetostrictive phase, or a polycrystal assembled from cy- lindrical magnetostrictive cubic crystals. The fibers could be aligned identically or randomly oriented in the transverse plane. There is no restriction on the cross-sectional shapes of the fibers, nor on the transverse arrangement of the compos- ite. The derivation is based on a construction of uniform fields,

^11,12

originally devised for finding effective thermal ex- pansions of two-phase composites and later generalized to polycrystals.

^13,14

We derive exact connections for three dif- ferent preferential orientations, with one of the crystallo- graphic axes

具

¹⁰⁰

典

^,

具

¹¹¹

典

^and

具

¹¹²

典

of the cubic magneto- strictive crystal being placed along the x

₃

direction. These results are theoretically rigorous and microstructurally inde- pendent. They are of particular value in that knowledge of the effective elastic moduli, either by theoretical analysis or through experimental measurement, will readily provide an estimate for the effective magnetostriction.

II. DERIVATION OF THE EXACT CONNECTIONS We consider a magnetostrictive crystal with cubic sym- metry, which corresponds to the most commonly reported

a兲Author to whom correspondence should be addressed; electronic mail:

[email protected]

491

0021-8979/2003/94(1)/491/5/$20.00 © 2003 American Institute of Physics

class with giant magnetostriction. The constitutive equation can be written as, based on the crystallographic coordinate x

⬘

␴

⬘⫽L⬘共

␧

⬘⫺␭⬘兲, 共1兲

where ␴

⬘

is the stress, ␧

⬘

is the strain and L

⬘

is the elastic moduli. The term

␭⬘

is the magnetostriction which depends on the magnetic field intensity H. As the crystal possesses cubic symmetry, the elastic moduli can be characterized by three independent moduli,

¹⁵

which are conveniently defined as L

₁₁⬘⬅a, L12⬘⬅b, L44⬘ ⬅c. Let us now consider a material

coordinate x

_i

which is related to the crystallographic axes x

_i⬘

by the transformation x

_i⫽

␣

i j

x

⬘_j

. For an external magnetic field H

₃

applied along the x

₃

axis of the material sample, the magnetostrictive strain along the crystallographic axes of a microcrystallite is given as

¹⁶

␭i j⬘⫽再

^{3 2 3 2}

^{␭␭100111冉}

^{␣ ␣}

³ⁱ³ⁱ²

^␣

^{⫺3 j}

^{, 1 3}

^冊

^{, for i for i}

^{⫽ j,⫽ j, 共2兲}

where

␭100

and

␭111

are the magnetostriction constants of the cubic microcrystallite. The constitutive Eq.

共1兲 can be recast

as

␴

⫽L共

␧

⫺␭兲, 共3兲

based on the material coordinate x, in which all field vari- ables are represented by unprimed quantities and their com- ponents follow the tensor transformation rules, L

_{i jkl}

⫽

␣

i p

␣

jq

␣

km

␣

ln

L

_pqmn⬘

^and

␭i j⫽

␣

i p

␣

jq␭pq⬘

. If the phase is isotropic, then L ÄL

⬘

and there exists a further reduction on the elastic constants, a

⫽b⫹2c. In the absence of body

force, the stresses must fulfill the divergence free equation.

At interfaces, perfect bonding conditions require that the dis- placement and traction be continuous across the interfaces. If the composite is statistically homogeneous, the aggregate can be regarded as a macroscopically homogeneous medium, whose constitutive equations, based on x, can be character- ized by similar relations

␴ ^¯

⫽L

*

_共

␧ ^¯

⫺␭

*

_{兲, 共4兲}

where the over bar denotes the volume average over the rep- resentative volume element and the asterisked quantities are the effective physical constants.

We first consider a two-phase particle reinforced com- posite consisting of inclusions and matrix. Both phases could be magnetostrictive. The constitutive relation of the inclu- sions is given by Eq.

共3兲 and the matrix follows the same

form, but with its field quantities distinguished by a subscript

‘‘0.’’ Crystallites are all identically aligned. We now separate the phases of the fiber and the matrix apart and apply a sufficiently large magnetic intensity field in the x

₃

axis, to reach the saturation state in both phases. This will cause a stress-free strain

␭ in the fiber and ␭0

in the matrix. Since the two quantities are in general different, the phases cannot be assembled together without inducing stresses. Let us now apply a constant strain ␧ ˆ through a homogeneous displace- ment condition on the boundaries of the fiber and of the matrix, and at the same time require that ␴

⫽

␴

0⬅

␴ ^{ˆ . This}

will automatically guarantee the satisfaction of stress equi- librium and displacement compatibility throughout the me- dium. By simple algebra it is found that

␧ ^ˆ Ä共L⫺L

0兲^⫺1共L␭⫺L0␭0兲. 共5兲

Now with the applied mechanical strain ␧ ˆ together with the phase magnetostriction, the whole composite medium has a uniform strain ␧ ˆ and stress ␴ ˆ throughout. This constitutes a particular set of uniform field solution, valid without any regard to the transverse microstructure of the composite. Use of ␧ ^{ˆ and} ␴ ^{ˆ in Eq.}

共4兲 will give an exact link between effec-

tive magnetostriction and effective moduli

L *

_␭

*

_{⫽L␭⫹共L}

*

_{⫺L兲共L⫺L0兲}^⫺1_{共L␭⫺L0␭0兲. 共6兲}

The exact correspondence

共6兲 is valid for two-phase compos-

ites, without any restriction on the geometric shapes of the particles nor on the elastic symmetry of constituent materi- als. It is noted that Eq.

共6兲 takes the same form as the well-

known exact correspondence between the effective thermal and mechanical effects found by Levin,

¹⁷

Benveniste and Dvorak,

¹⁸

and Milton,

¹⁹

among others. This is due to our assumption that the reinforced particles are all aligned iden- tically, and thus each inclusion has a constant magnetostric- tion with respect to x, similar to the thermal expansion tensor.

For technological applications the assumption that all magnetostrictive crystallites are all identically aligned is quite restrictive, and indeed not very realistic. To explore its potential for wider applications we now consider two com- posite systems:

共i兲 two-dimensional polycrystals made of cy-

lindrical magnetostrictive cubic crystallites and

共ii兲 two-

phase fibrous

共or rod-like兲 composites consisting of

magnetostrictive fibers and isotropic matrix. The fibers, with one of the preferential crystallographic axes being aligned in the x

₃

axis, are randomly oriented in the transverse x

₁⫺x2

plane. The transverse cross sections of the fibers could be arbitrary in shape. Thus on a macroscopic scale the hetero- geneous medium is effectively transversely isotropic. Again, we start from the constitutive Eq.

共3兲 based on the material

axes x, in which the components of L and

␭ can be written in

terms of a,b,c,

␭100

,

␭111

. Let us now separate each fiber and the matrix apart and apply a sufficiently large magnetic intensity field in the x

₃

direction to attain the saturation state.

Now since each fiber has its own orientation with respect to the x

₃

axis, its magnetostriction is generally a function of one rotation angle versus the x

₃

axis, say ␪ . This will cause a stress-free magnetostriction strain

␭(

␪ ) in each fiber and

␭⬘

in the matrix. Apparently, they are not compatible and cannot be assembled together. Since the fibers are arbitrarily oriented, to ensure the conditions of stress equilibrium and compatibility conditions at the interfaces, the fields need to be hydrostatic in the transverse plane. Thus, we now apply a homogeneous strain field ␧ ^ˆ

⫽关␧ˆ1

,

␧ˆ1

,

␧ˆ3

,0,0,0

兴, in which

␧1⫽␧2⫽␧1 0⫽␧2

0⫽␧ˆ1

,

␧3⫽␧3 0⫽␧ˆ3

,

␧4⫽␧5⫽␧6⫽␧4 共7兲

0⫽␧5 0⫽␧6

0⫽0.

In the same time, we request that

␴

1⫽

␴

2⫽

␴

1 0⫽

␴

2

0⫽

␴ ^ˆ

1

,

␴

4⫽

␴

5⫽

␴

6⫽

␴

4 共8兲

0⫽

␴

5 0⫽

␴

6

0⫽0

throughout the composite. Satisfaction of Eqs.

共7兲 and 共8兲

will constitute an admissible uniform field solution for the considered system. Evidently, these uniform fields are ex- actly the volume average field quantities in Eq.

共4兲. The com-

ponent ␴ ^¯

3

simple follows ␯␴

3⫹

␯

0

␴

3

0

, where ␯ ^and ␯

0

are the volume fraction of the fiber and the matrix, respectively.

For two-dimensional polycrystals, the quantities associated with the matrix phase in Eqs.

共7兲 and 共8兲 are taken out and

␴

¯

3

simple follows ␴

3

. Upon substituting the uniform field solutions in Eq.

共4兲, one will find the exact connections be-

tween

␭

* and L * and, in some cases, we also construct exact connections between the components of L * . One of the fa- mous known results of the latter kind is the universal con- nection found by Hill,

²⁰

in which he showed that among the five effective constants of an effective transversely isotropic fibrous composite only three of them are independent. See also Ref. 21 for a unified generalization to a broader context.

We now give exact connections for three different cases, with one of the crystallographic directions

具

¹⁰⁰

典

^,

具

¹¹¹

典

^and

具

¹¹²

典

of the cubic crystals being placed along the x

₃

direc- tion. For simplicity, we define the following short notations:

⌬

␷

⬅

␷

⫺

␷

0

and

具

␷

典^⬅

␯ ␷

⫹

␯

0

␷

0

.

共A兲具

¹⁰⁰

典

along the x

₃

axis. We find that the components of L * are not all independent, which are necessarily con- nected by

␣

1共L11

*

_⫹L12

*

_兲⫺L13

*

_⫽

␣

1共a⫹b兲⫺b,

2 ␣

1

L

₁₃

*

_⫺L33

*

_⫽2

␣

1具

^b

典⫺具

^a

典

^.

共9兲

Further

␭

* can be directly linked with L * through

冋^共L11

^{* 2L}

^⫹L13

^*

¹²

^*

^{兲 L}

^L

¹³

^*

³³

^*

册冋^␭1

^*

^␭⫹3

^{* ␣}

²册^⫽冋

^{␣ 2}

^{2共a⫹b兲⫺␭}

^␣

^2具

^b

^典⫹

^␯

^{␭100100共a⫺b兲/2共a⫺b兲 册}

^,

共10兲

where

␣

1⫽ ⌬b

⌬共a⫹b兲

, ␣

2⫽␭100共a⫺b兲

2

⌬共a⫹b兲

.

共11兲

If the matrix is absent and the aggregate becomes a two- dimensional polycrystal, the following exact results can be identified: L

₁₁

*

_⫹L12

*

_{⫽a⫹b, L13}

*

_{⫽b, L33}

*

_{⫽a, ␭1}

*

_⫽␭2

*

⫽⫺␭100

/2,

␭3

*

_⫽␭100

.

共B兲具

¹¹¹

典

along the x

₃

axis. We find that

␣

3共L11

*

_⫹L12

*

_兲⫺L13

*

_⫽²₃

␣

3共a⫹2b⫹c兲⫺¹3共a⫹2b⫺2c兲,

2 ␣

3

L

₁₃

*

_⫺L33

*

_⫽²₃

␣

3具

^a

⫹2b⫺2c典⫺¹3具

^a

⫹2b⫹4c典

^,

共12兲

冋^共L11

^{* 2L}

^⫹L13

^*

¹²

^*

^{兲 L}

^L

¹³

^*

³³

^*

册冋^␭1

^*

^⫹3␭3

^{* ␣}

⁴册

⫽冋

^{2 ␣}

⁴共a⫹2b⫹c兲⫺c␭111

2 ␣

4具

^a

⫹2b⫺2c典⫹2

␯ ^c

␭111册

^,

^共13兲

where

␣

3⫽ ⌬共a⫹2b⫺2c兲

⌬共2a⫹4b⫹2c兲

, ␣

4⫽

c

␭111

⌬共2a⫹4b⫹2c兲

.

共14兲

For two-dimensional polycrystals, namely ␯

⫽1, exact

moduli are determined as

␭1

*

_⫽⫺␭111

/2,

␭3

*

_⫽␭111

, L

₁₁

*

⫹L12

*

_⫽²₃

(a

⫹2b⫹c), L13

*

_⫽¹₃

(a

⫹2b⫺2c), L33

*

_⫽¹₃

(a

⫹2b

⫹4c).

共C兲 具

¹¹²

典

along the x

₃

axis. Only the magneto- mechanical correspondence can be found

冋^共L11

^{* 2L}

^⫹L13

^*

¹²

^*

^{兲 L}

^L

¹³

^*

³³

^*

册冋^{共␭共␭1}

^*

³

^*

^⫹⫺

^{␣ ␣}

^56兲兲册

⫽冋^冉

^{2 3 a}

^⫹冓

^{a 2 4 3}

^⫹

^b

^⫹

^{3b 2 2 3}

^⫺c

^c

^冊

^␣

冔⁵

^␣

^⫺5⫺冉

^{1 3}

冓

^{a a 2}

^⫹⫹

^{2 3 b 2 b}

^⫹c⫺

^{2 3}

冔

^{c ␣}

^冊6

^␣

^⫹6⫺c␭

^␤

⁰

^␯

¹¹¹

^,

册

^,

共15兲

where

␣

5⫽

p 2

⌬共a⫹2b⫺2c兲

⌬共a⫹2b兲 ␭¹⁰⁰⫹q ⌬b

⌬共a⫹2b兲 ␭¹¹¹

,

共16兲

␣

6⫽p⌬共a⫹2b⫹c兲

⌬共a⫹2b兲 ␭¹⁰⁰⫹q ⌬共a⫹b兲

⌬共a⫹2b兲 ␭¹¹¹

and ␤

0⫽␭100

(a

⫺b)/4⫹3c␭111

/2, p

⫽(a⫺b)/(a⫺b⫺2c),

q

⫽⫺3c/(a⫺b⫺2c).

For two-dimensional polycrystals, we have L

₁₁

*

_⫹L12

*

⫹L13

*

_⫽2L13

*

_⫹L33

*

_{⫽a⫹2b, and}

冋

^L

¹¹

^{* 2L}

^⫹L13

^*

¹²

^{* L L}

¹³

^*

³³

^*

册冋^␭1

^*

^␭⫹3

^{* ␰}

册

⫽冋

^p

^{共a⫹2b⫹c兲␭}

^p

^{共a⫹2b⫺2c兲␭100⫹q共a⫹b兲␭100⫹2bq␭111111}册

^,

^共17兲

where ␰

⫽³2

p

␭100⫹q␭111

.

In all three cases the effective magnetostriction of the composites can be exactly calculated once the effective moduli of L

₁₁

*

_⫹L12

* , L

₁₃

* , L

₃₃

* are known. This linkage is theoretically exact and is independent of the micromechani- cal models used to evaluate the effective moduli. For ex- ample, if one uses Voigt’s estimate to calculate L * , our Eqs.

共10兲, 共13兲 and 共15兲 will give exactly Voigt’s effective ␭

* which was previously formulated as

²²

␭Voigt

*

_⫽

␯

共

␯

具

^L

典^orien⫹

␯

0

L

₀兲^⫺1具

^L

^␭典^orien

^.

^共18兲

Here

具

^L

典orien

means the orientational average of all crystal- lites about the axial axis. Likewise, if one uses Mori–Tanaka or self-consistent method to calculate L * , our

共10兲, 共13兲 and 共15兲 will give, respectively, MT and self-consistent ␭

* . In fact, an experimental measurement of these moduli L

₁₁

*

⫹L12

* , L

₁₃

* , L

₃₃

* will readily provide a quick estimate for

␭

* .

Note, however, that in the construction of uniform fields we

have enforced that the axial strain in the crystallites be the

same with that of the matrix to ensure the required axial

compatibility for fibrous aggregates. Thus, the Reuss model,

which assumes constant stress, apparently violates this re-

quirement and should not be adopted in our connections.

Any other micromechanical models, which employ Green’s function and/or Eshelby’s tensors, as well as Voigt’s model, will automatically fulfill the equal axial strain requirement and thus will provide reasonable estimates for

␭

* through Eqs.

共10兲, 共13兲 and 共15兲.

III. RESULTS AND DISCUSSION

To illustrate our results, we calculate the effective mag- netostriction for fibrous composite systems made of Terfenol-D magnetostrictive crystallites and epoxy matrix.

When the volume fraction of the magnetostrictive crystallites

␯ equals one, the calculated value gives the effective mag- netostriction for two-dimensional polycrystals. The properties

⁸

of the constituent phases used for calculations are a

⫽101 GPa, b⫽40 GPa, c⫽38 GPa, ␭111⫽1700 ppm,

␭100⫽100 ppm for Terfenol-D, and a0⫽6.5 GPa, b0⫽3.5

GPa, c

₀⫽1.5 GPa for epoxy. The effective elastic moduli of

composites and polycrystals are evaluated based on two simple models: Voigt’s constant strain model and a modified Voigt’s assumption. Here the modified Voigt’s model as- sumes that a constant axial strain prevails in the composite, while in the remaining directions the stress components are uniform throughout. Figures 1

共a兲–1共c兲 show the effective en-

gineering magnetostriction

␭s⫽²3

(

␭3

*

_⫺␭1

* ) for three differ- ent cases

共A兲, 共B兲 and 共C兲. We have correctly verified that

our calculations based on Eqs.

共10兲, 共13兲 and 共15兲 using Voi-

gt’s effective moduli are exactly the same as those directly calculated from Eq.

共18兲. We have also verified analytically

that, when ␯ →1, the effective magnetostrictions from Eqs.

共10兲 and 共13兲 recover the exact results for 具

¹⁰⁰

典

^and

具

¹¹¹

典

polycrystals. The latter effect can be observed in the Figs.

1

共a兲 and 1共b兲 that the Voigt and modified Voigt curves coin-

cide with each other at ␯

⫽1. Also we remark the result of

Eq.

共15兲 yields the correct value of Eq. 共17兲 when

␯

⫽1. Note

that in Fig. 1

共c兲 at

␯

⫽1, ␭s

takes slightly different values for the two estimates. This is due to the fact that the two models give different effective elastic constants for case

共C兲.

In conclusion, we have derived exact connections be- tween the effective magnetostriction and the effective elastic moduli for fibrous composite systems and two-dimensional polycrystals. The magnetostrictive crystallites are cubic sym- metry with one of the directions

具

¹⁰⁰

典

^,

具

¹¹¹

典

^and

具

¹¹²

典

being aligned in the x

₃

axis. These results are of theoretical and technological value in that knowledge of the effective elastic moduli readily provides an estimate for the effective magnetostriction of the composite medium.

ACKNOWLEDGMENTS

This work was initiated while T.C. was a visitor at the Department of Engineering Mechanics, Tsinghua University.

Financial support from NSC 91-2211-E006-085, Taiwan, and from NSF, China, are gratefully acknowledged. C.W.N. ac- knowledges the support of the Ministry of Sciences and Technology of China under Grant No. G2000067108 and G.J.W. was supported by the National Science Foundation, under Grant No. CMS 01-14801.

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