ZorskiAMS Micunovic Revised for Reviewer

8/3/2019 ZorskiAMS Micunovic Revised for Reviewer

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Thermodynamical and self consistent

approach to inelastic ferromagneticpolycrystals 0

M. MICUNOVIC

Faculty of Mechanical Engineering, University of Kragujevac,Sestre Janjica 6a, 34000 Kragujevac, Serbia and Montenegroe-mail: [email protected]

Dedicated to memory of Professor Henryk Zorski

with gratitude and admiration

Geometric and kinematic aspects of intragranular as well as intergranular plas-tic deformation of ferromagnetic polycrystals are discussed. Elastic strain iscovered by the effective field homogenization method inside a representativevolume element (RVE). By applying this method an effective magnetostriction4-tensor is determined. Evolution equation formed by tensor representationhaving incremental form is postulated to model inelastic metals. The rate de-pendence takes place by means of stress rate dependent value of the initialyield stress. Concept of M. Zorawski deformation geometry is extended on thebase of constrained micro and free macro rotations in intermediate referenceconfiguration. This has as a consequence that evolution equation for plastic

spin of RVE is an outcome of evolution equation for plastic stretching. Themacroscopic evolution equation is based on the Vakulenko’s concept of ther-modynamic time. A tensor representation for magnetomechanical interactionis proposed and susceptibility coefficients for iron are calibrated.Keywords: Anisotropic Eshelby tensor, Tensor representation, Vakulenko’sthermodynamic time, Susceptibility coefficients.

1 Introduction

The principal objective of this paper is to give a simplified approach to in-elasticity of (inherently polycrystalline) ferromagnetic materials aimed to serveprimarily to nondestructive magnetic examination of inelastic behavior of reac-tor steels (cf. [30, 56, 47]). The subject is complicated and requires a carefulexamination of geometry as well as thermodynamics of deformation process.

As the starting point the geometry of deformation with multiplicative de-formation gradient must be extended in order to include magneto-mechanicalinteraction. This is done in the easiest way following the Kroner’s “incom-patibility method ” which in the paper [15] was applied to magnetostriction.

0Submitted to Archives of Mechanics, Warszawa 2005

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The authors assumed that total incompatibility composed of purely elastic and

magnetostrictive strain is zero while its constituents are not. They assumedisoclinicity of magnetization vectors in the natural state elements (intermedi-ate local reference configuration) and their inhomogeneity in instant deformedconfiguration which is responsible for magnetostrictive strains. Such an as-sumption is very much in accord with our geometrical approach (cf. [32]) andis accepted henceforth. An extension of their reasoning to the more generalcase of thermo-elasto-magneto-plastic strain is allowed if Kroner’s formula

(1) F = FE FP

is understood as follows. The incompatible tensor FE is obtained by cutting

the body into infinitesimal pieces which are free of neighbors, brought to thereference temperature in the absence of external magnetic field. The otherconstituent on right hand side of (1), namely FP , contains strain caused byirreversible magnetization as well as by pure plastic strain. This will be furtherelaborated in the next section.

Our approach to evolution equations does not take as granted associativ-ity of flow rule i.e. the normality of the plastic stretching tensor onto a yieldsurface [34, 35, 28] even if such an approach is accepted in the majority of thepapers dealing with the subject. Such a normality is seriously questioned notonly by the theoretical but by experimental results as well.1 For these reasonsthe normality is at first abandoned and instead of such an assumption evolu-

tion equations (exposed in the second subsection of this section) are based onthe appropriate geometry of deformation and tensor representation. Each rea-sonable constitutive theory must have a tight relationship to thermodynamics.Here we mention some possible approaches.

• A very attractive approach to the extended thermodynamics has been pro-posed in [42] with a rational analysis of thermodynamic processes leadingto the desired thermodynamic restrictions of general constitutive equa-tions. This approach with the Liu’s theorem [25] was applied by thisauthor to viscoplastic materials in [34] and to inelastic composite materi-als in [35]. In spite of its beauty an inherent coldness function (which isnot quite clear from the experimental point of view) is inevitable.

• An alternative approach to extended thermodynamics following [8] wasapplied to thermoplasticity of irradiated materials in [28] and viscoplas-ticity of single ferromagnetic crystals in [37].

1In the paper [16] a comparison between tension and torsion was one of the first signs of such a discrepancy. This subject has been discussed in detail in [41] where also experimentsdealing with cruciform specimens [1] are included. In so called J 2-theories with corners wherea lot of “normals” to yield surface exist this normality is in fact abandoned.

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• The approach of endochronic thermodynamics with properly chosen ther-

modynamic time is the succeeding choice. It will be presented here in theversion of Vakulenko [53].

• Statistical thermodynamics is a mighty tool in treating choice of internalvariables. It will be partly used here in defining magnetization distribu-tions. In this field developments of Zorski [57] and Kroner [22] are veryimportant.

The analysis in this paper is aimed to a description of fast multiaxial ex-periments on austenitic steels like AISI 316H having face centered cubic lattice(compare [36]) as well other steels with body centered cubic lattice. For thissake it is essential to reduce the number of material constants to be found from

the available experiments. In other words, the general desire is always to makeevolution equations with minimal number of material constants even if theseequations originate form very general functionals. The evolution equations usu-ally comprise of plastic stretching (often named by experimentalists as plasticstrain rate tensor) as well as plastic spin. The later is understood by someauthors as a trigger in localization behavior while some others require indepen-dence of these two evolution equations which greatly complicates identificationproblem. This issue will be touched as well.

Means applied in the paper to realize the stated goal are listed below. Micro-evolution equation having incremental form is postulated. The rate dependencetakes place by means of stress rate dependent value of the initial yield stress.

Thus, such materials are quasi-rate dependent. Such an approach follows re-sults of dynamic experiments on AISI-steels performed at JRC-Ispra, Italy atstrain rates in the interval [0.001, 1000] 1/s (described in [36]. The still con-troversial plastic spin issue is treated by extending the concept of M. Zorawskideformation geometry postulating constrained micro and free macro rotationsin intermediate reference configuration. The self-consistent method (effectivefield variant) resulting from paper [24] provides effective stiffness tensors forRVE from individual grains and leads to a simplification of tedious calculationsof their effective stiffness fourth rank tensors of individual grains.

Throughout the paper thermal, elastic and magnetostrictive strains are as-sumed to be small being approximately additive. This does not hold for plastic

strains which are finite in general. However, in the fourth section devoted tothe self consistent method plastic strains are also considered as small due toassumptions in “effective field theory ” developed in the papers [18, 19, 24].

At present, only first deformation gradients are considered. Thus, higherorder effects like mechanical couple stresses (requiring a gradient viscoplasticitytreated in [39]) as well as phase transitions, ferroelectric and ferrimagneticeffects, intrinsic spin, exchange forces and gyromagnetic effects are ignored(with negligible precessional velocity of magnetization). The considered process

3



is electromagnetically slow enough such that ratio of particle velocity and speed

of light is negligible. Due to size of paper boundary conditions are not analyzed.The notation used in this paper might be briefly shown by summation over

repeated Cartesian indices:−→H

−→M = H a M a, (

−→H ⊗

−→M )ab = H a M b, (AB)ab =

Aac Bcb, tr(AB) = Aac Bca, (DE)ab = DabcdE dc. The usual notation: 2 symA =A + AT and 2 skwA = A − AT is also applied here. Due to limited space somesecond order effects (treated in detail in [39]) have to be dropped from the con-sideration. For the same reason many important references cannot be includedinto already long list of references.

2 Micro and macro-geometry

2.1 Magneto-thermo-mechanical distortions

As a prerequisite, a correct geometric description of an inelastic deformationprocess analyzed is necessary. Consider a polycrystalline body in a real con-

figuration (k) ≡ (k(t)) with dislocations, magnetization−→M (X, t) and an inho-

mogeneous temperature field T (X, t) (where t stands for time and X for theconsidered particle of the body) subject to external stress (i.e. surface tractions)

and external magnetic field−→H . Corresponding to (k) there exists, usually, an

initial reference configuration (K ) = (k (t0)) with dislocations at a homoge-neous temperature T 0 without surface tractions. Due to these defects such aconfiguration is not stress-free but contains an equilibrated residual stress (so

called “back-stress”). It is generally accepted that linear mapping functionF(., t) : (K ) → (k) is compatible second rank total deformation gradient ten-

sor. In the papers dealing with continuum representations of dislocation distri-butions configuration (k) is imagined to be cut into small elements denoted by(n) ≡ (n(t)), these being subsequently brought to the temperature of (K ) freeof neighbors. The deformation “gradient” tensor FE (., t) : (n) → (k) obtainedin such a way is incompatible and should be called the thermoelastic distortiontensor whereas (n)-elements are commonly named as natural state local refer-ence configurations (cf. for instance [21, 32, 15]). Of course, the correspondingplastic distortion tensor is not compatible. Here F is found by comparison of material fibres in (K ) and (k) while FE is determined by crystallographic vec-

tors in (n) and (k). Multiplying (1) from the right hand side by FE (., t0)−1

we reach at slightly modified Kroner’s decomposition rule 2. The formula (1)is often wrongly named as Lee’s decomposition formula. It is worthy of notethat curlFE (., t)−1 = 0 and this incompatibility is commonly attributed to an

2This modification, representing mapping (k(t0)) → (k(t)) and introduced by Teodosiu[51] is necessary to account for Dashner replacement invariance [12].

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asymmetric second order tensor of dislocation density3.

Taking into account the above discussion, it is reasonable to decomposeirreversible magneto-plastic distortion as follows:

(2) FP = FRµ F p.

It contains irreversible residual magnetic distortion FRµ and pure plastic distor-

tion F p. On the other hand, magneto-thermo-elastic distortion may be decom-posed by means of:

(3) FE = FeFθFrµ

with pure elastic distortion Fe, thermal distortion Fθ as well as reversible mag-

netic distortion Frµ. Now it is reasonable to define thermo-magnetic quasi-plastic

distortion tensor as the product of thermal and magnetic distortion:

(4) Fω = FθFrµFR

µ .

The name “quasi-plastic” was introduced by Anthony in [2]. Suppose nowthat thermal as well as magnetostrictive strains are much smaller than plasticstrain. This is confirmed by experiments even for plastic strains of the orderof 1%. Then for thermo-magnetic quasi-plastic strain a linear decompositionapproximately holds:

(5) Eω ≈ Eθ + Er

µ + ER

µ ≡ Eθ + Eµ.

Here Eα = (FT αFα − 1)/2, α ∈ ω ,θ,µ are Lagrangian strains and Eµ is called

magnetostrictive strain . Due to magnetic symmetry it is bilinear function of magnetization vector [31]. Its constituents are reversible magnetostrictive strain and irreversible magnetostrictive strain . Their explicit forms are given in thenext section.

2.2 Polycrystal strains

Let us imagine now that a typical (n)-element (called in the sequel represen-tative volume element and denoted by RVE) is composed of N g single crystalgrains such that each Λ-th grain has N s slip systems AαΛ = −→s αΛ ⊗ −→n αΛ. Forinstance, for fcc-crystals N s = 24. Here −→s αΛ is the unit slip vector and −→n αΛ

3This definition covers only geometrically necessary dislocations (GND) without statisti-cally stored dislocations (SSD) appearing at dislocation loops and dipoles. For this reason inthe paper [22] a more precise definition is given by infinite number of correlation functionscomposed by fundamental dyadic of Burgers vector and dislocation line tangent vector. Onthe other hand, FP and its first gradients allow diverse non-Euclidean interpretations coveringnot only GND but implanting Eshelbian strains as well (cf. details in [32, 40]).

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is the unit vector normal to the slip plane. Comparing a RVE in (n(t)) and

(n(t0)) we may write a formula similar to Kroner’s formula (1) holding for themicroplastic distortion tensor

(6) ΠΛ := ΠΛE ΠΛP ,

whose components are the residual microelastic distortion tensor ΠΛE and mi-croplastic distortion tensor ΠΛP . Then the polar decomposition gives

ΠΛE = RΛUΛE .

Here microrotation satisfies the relations RT ΛRΛ = 1 and its time rate equals to

DtRΛ

= ΩΛ

RΛ

. Therefore, slip systems dyadics evolve according to: AαΛ

(t) =RΛ(t)AαΛ(t0)RT

Λ(t) as microrotations must be constrained inside each RVE. Bymaking use of these dyadics as well as microrotations we may write

UΛE = diag(1 + λkΛ), k ∈ 1, 2, 3

as well as

ΠΛP := 1 +α

γ αΛAαΛ,

where γ αΛ (α ∈ 1, N s , Λ ∈ 1, N g) are plastic microshears inside the Λ–thgrain. If a RVE has the volume ∆V = Λ ∆V Λ and the microplastic deforma-

tion tensors for individual grains are

CΠΛ = ΠT ΛP U

2ΛE ΠΛP ≡

1 +

α

γ αΛAT αΛ

U2

ΛE

1 +

α

γ αΛAαΛ

,

then their volume average named macroplastic deformation tensor CP := FT P FP

has the following form:

(7) CP = CΠΛ =

ΠT ΛΠΛ

≡

1

∆V

Λ

ΠT ΛΠΛ∆V Λ.

Moreover, in the corresponding polar macro-decomposition FP = RP UP the macroplastic rotation tensor RP is free to be taken arbitrary (according toZorawski [58] and might be fixed either to be a unit tensor or to have Mandel’sisoclinicity property (details are given in [41]). For a definition of isoclinicitywe should have to find average crystal directions in RVE(t) and RVE(t0) andto make them coincident. Accepting henceforth the first choice we acquire therelationship

(8) RP = 1 ⇒ FP = UP = C1/2P ,

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0RVE(t)0

RVE(t)

F (t) p U (t) p=

(K)

(t)0FE

(t)FE

(t)F

(t))(k

(n)0

(n(t))

Figure 1: Principal configurations of a polycrystal body with illustration of freemacro and constrained micro-rotation

which greatly simplifies macroplastic spin issue (cf. [41]).The above introduced microrotations of grains permit the exact relationship

for material time rate of microplastic distortion tensor (cf. figure 1)

DtΠΛP =α

AαΛDtγ αΛ + γ αΛDtAαΛ.

Here the aforementioned constrained microrotations must fulfil the relationship

DtAαΛ = ΩΛAαΛ + AαΛΩT Λ,

such that microplastic stretching and microplastic spin tensors read:

DΛΠ = RT Λ (DΛP + Dt log UΛE ) RΛ, WΛΠ = RT

ΛWΛP RΛ + WΛE ,

withWΛE = DtR

T ΛRΛ

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and

Dt log UΛE = diagDtλkΛ (1 + λkΛ)−1 , k ∈ 1, 2, 3

as well as next notations

2DΛP = DtΠΛP Π−1ΛP + Π−T

ΛP DtΠT ΛP , 2WΛP = DtΠΛP Π

−1ΛP − Π−T

ΛP DtΠT ΛP .

The corresponding macroplastic stretching and macroplastic spin tensors follownow directly from (8) in the following form:

(9) 2DP = DtUP U−1P + U−1

P DtUP , 2WP = DtUP U−1P − U−1

P DtUP .

It is worthy of note that such a representation considerably reduces num-

ber of necessary material constants if some evolution equations for macro-quantities DP and WP are chosen in such a way to follow from tensor rep-resentation. Connection of the macroplastic stretching with (7) by means of 2DP = U−1

P DtCP U−1P is then straightforward and is obtained from:

DtCΠΛ = ΠT ΛP DtU

2ΛE ΠΛP +

α

AαΛDtγ αΛ + γ αΛDtAαΛ

U2

ΛE ΠΛP +

ΠT ΛP DtU

2ΛE

α

AT αΛDtγ αΛ + γ αΛDtA

T αΛ

by means of the spatial averaging throughout a RVE i.e. DtCP = DtCΠΛ .

In the paper [38] an initial attempt has been made to model transition of plastic strain from a grain to its neighbors. Yet an application of such an ideato computer simulation of inelastic behavior of RVE would require too largecomputing time. Instead of that here a self-consistent method is applied. Itwill be explained in more detail in the subsequent section.

3 Macroscopic evolution and constitutive equations

3.1 Evolution equations by extended thermodynamics

Let us consider only magnetomechanical terms in balance laws and constitutive

equations. More precisely the scope of this subsection is limited by the nextassumption:

Assumption 3.1 Ferroelectric and ferrimagnetic effects, intrinsic spin, ex-change forces and gyromagnetic effects are ignored (with negligible precessional velocity of magnetization). The considered process is slow enough such that ratio of particle velocity and speed of light is negligible.4

4Details of such a situation are explained in [31, 37].

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Then, the next reduced set of objective and Galilei-invariant state variables

[30, 46] should be introduced in general

(10) Γ := E, EP ,A,T,GRADT, Q, M, MR, Γ ∈ G,

The tensorial quantities are chosen here in such a way to be connected to theconvective material X -coordinates5 in the deformed instant (k)-configuration(cf. figure 1).Herein

2 E = FT F − 1 ≡ C − 1 is the Lagrangean total strain tensor,

2EP = FT P FP − 1 ≡ CP − 1 - Lagrangean plastic strain tensor,

GRADT ≡ F−1gradT - temperature gradient,

Q ≡ J F−1 q - the heat flux vector ,

M = J F−1 m - the magnetization vector,

MR - the corresponding irreversible (residual) magnetization vector,

A - the volume defined dislocation density (number of dislocationlines per unit volume).

Capital letters are reserved for such a convective representation. Differential

operator GRAD ≡−→C K ∂ K ⊗ is referred to such coordinate frame, whereas

grad ≡ −→g a∂ a⊗ is used to indicate the same operator in spatial (possibly Carte-sian) coordinate frame of (k)-configuration. Accordingly, GRADT, Q, M, MR

have convective material either covariant or contravariant components. Theabove set Γ may be otherwise understood as a point belonging to the ex-tended configuration (deformation-temperature-magnetic) space G. Its subsetEP , A, MR collects internal variables responsible for irreversible behavior.

To this configuration point there corresponds a reaction point representedby the set

(11) ∆1 := TK ,u,s, S, E , B, ∆1 ∈ D,

where6

5Another choice is to accept convective structural coordinates in one of local natural state

configurations depicted on the figure 1. However, this is more difficult here since it is necessaryto account for non-Euclidean expressions of differential operators (cf. [41] for details).

6Our convective magnetic vectors coincide with those in the comprehensive reference([31, p. 169]) with the exception of magnetic induction and magnetization where M =J C−1 MMaugin and B = J −1 C BMaugin holds.

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TK = J F−1TkF−T - the symmetric Piola-Kirchhoff stress tensor of

the second kind related to the material convective coordinates of (k)-configuration wherein Tk is the Cauchy stress,

u and s - the internal energy and the entropy densities,

S ≡ J F−1s - the entropy flux vector,

B = bF - the magnetic induction vector.

By means of D the extended stress space is indicated, whose objective andGalilei-invariant elements are listed hereinabove in (11). At this place theconstitutive equations are simply stated by the bijective mapping:

(12) ∆1 = R(Γ) ≡ ∆1(Γ) or R : G → D ,

which is too general so that the thermodynamic analysis presented henceforth isaimed to supply restrictions concordant with the second law of thermodynamics.

The evolution functions are proposed here in such a way to be compatiblewith (10)-(12) and are collected into the set

(13) ∆2 := Q∗, E∗, M ∗, A∗, ∆2 ∈ D,

such that objective evolution equations simply read:

(14) Dt Q = Q∗(Γ),

(15) DtEP = E∗(Γ),

(16) Dt MR = M ∗(Γ),

(17) Dt A = A∗(Γ),

where the material time derivative is designated by Dt. The simplicity of lefthand sides of (14-17) owing to the absence of corotational time derivativeshas the origin in the accepted material convective description of constitutivefunctions and variables listed in (10)-(11)7.

A thermodynamic process occurring in the considered body is described bythe evolution equations and by the following balance laws which are equivalent

but slightly modified with regard to those of [31]:

ρDtu −

TK +

1 ( B M) − B ⊗ M

C−1

: DtE −

BDt M + DIV Q − ρ0h = 0,

(18)

7It is noted here that right hand sides of (14-17) do not include material time derivativesof internal variables - elements of the set ∆1 in (11). In other words, quasi rate independentferromagnetics (cf. [40]) are not covered by the above evolution equations (14)-(17). In thesubsections 3.2 and 3.4 this approach is extended to include such materials as well.

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(19) ρ0 − ρJ = 0,

(20) ρDtv − f − f em −1

J DIV (TK F

T ) = 0,

(21) skwTK = skw

C−1( B ⊗ M − P ⊗ E )

,

wherein ρ0 and ρ are mass densities in (K ) and (k), v is the velocity of theparticle, while the conventional notation:

J f emF ≡ J GRAD(1

J BFT )F−T M,

is used. The above equations (18)-(21) are, respectively, the equation of energybalance, the mass conservation law, the equation of balance of momentum andthe equation of balance of angular momentum.

Let us remind that electric quantities are not considered in this paper. Thenone of Maxwell equations becomes identity whilst the others read:

(22) Dt B = 0,

(23) DI V B = 0,

(24) CURL( B − J −1C M) = 0.

Further consequence of the assumption 3.1 is a simplification of the set of internal variables loosing from it gradient of the magnetization vector assumingin such a way that balance law for magnetization [45] (i.e. balance of angularmomentum of spin continuum in wording of [31]) is identically satisfied.

The above listed balance laws imply constraints on the elements of the setΓ ∪ DtΓ causing breaking of their independence which is the essence of theLiu’s theorem (given in [25]).

Nonetheless, there is still another constraint on these elements in the case of inelastic deformation process: the essential notion of yield surface which dividessharply two regions of material behavior. In other words, the elastic and plastic

strain ranges are separated by the yield surface.8 The dynamic and static scalaryield functions are here defined by means of

(25) f = f (TK , T, EP , MR) ≡ h(Γ),

8An essentially important question to be posed here reads: whether such a frontier betweenreversible and irreversible magnetization exists or not. If the answer is affirmative, then itleads to next questions: which irreversible process (mechanical or magnetic) advances andwhat is shape of magnetic “yield” surface? It is tacitly assumed in this subsection that bothprocesses are simultaneously triggered. In subsection 3.4 some other possibilities are discussed.

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(26) f # = f (T#K , T, EP , MR) ≡ h0(Γ),

where T#K is static stress corresponding to the dynamic viscoplastic stress TK .

Their difference is usually termed as the overstress tensor and in the simplestcase it may be represented by a linear function of DtEP as follows:

(27) ∆TK := TK − T#K = P(Γ) : DtEP ,

with P(Γ) being fourth rank tensor of plastic viscosity coefficients. Introducingthe plastic strain rate intensity by

(28) Dt p := (DtEP : DtEP )1/2 ≡ DtEP ≥ 0,

the classification:f > 0, f # = 0, Dp > 0 - viscoplastic behavior;f = f # = 0, Dp = 0 - elastoplastic frontier;f = f # < 0, Dp = 0 - elastic behavior;

and the kinematic constraint:9

(29) < f > Dtf # = 0,

will be useful in the sequel.All thermodynamic processes must obey the master law of nature i.e. the

second law of thermodynamics which in material convective coordinate frame

of (k)-configuration reads:

(30) ρDts + DIV S − ρr

T = 0,

where r/T is the entropy source. Precisely defined a thermodynamic process isa solution of evolution and balance equations which obeys (30). The analysisof the above entropy inequality (30) by the Liu’s theorem may be describedas follows. Replacing s∗(Γ) and S ∗(Γ) into (30) this becomes a differentialinequality linear with respect to the elements of the set DtΓ ∪ GRADΓnamely:

9Here < f >= 1 for f > 0 and f = 0 otherwise. This notation should not be confused

with averaging sign < • > used in the previous section.

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ρ0Dts + DI V S − ρ rT

− Λu ρ0Dtu − BDt M + DIV Q − ρ0h−

DtE :

TK +

1 ( B M) − B ⊗ M

C−1

−

Λ : skw

TK − C−1 B ⊗ M

− Λf < f > Dtf #−

Λv

ρDtv − f − f em − J −1DIV (TK FT )

− ΛA

DtA − A∗(Γ)

−

ΛE :

DtEP − E∗(Γ)

− ΛM

Dt MR − M ∗(Γ)

−

Λ2 Dt B − Λ3 DI V B − Λ4 CURL( B − J −1C M) ≥ 0.

(31)

By introducing Lagrange multipliers into (30) all the elements of the setDtΓ ∪ GRADΓ except solely GRADT (which is already included into Γ)become mutually independent. Hence, in thus extended inequality all the co-efficients with the elements of the set DtΓ ∪ GRADΓ must vanish. Thisgives rise to the consecutive constitutive restrictions (cf. [28]):

(32) S = Λu(T ) Q ≡ Q/T,

(33) TK = C−1

− M ⊗ B + ( B M)1

+ ρ0∂ Eg+ < f > T Λf ∂ Ef #

,

(34) s = ∂ T g+ < f > Tρ−1Λf ∂ T f #,

(35) B = −ρ0∂ Mg− < f > T Λf ∂ Mf #,

(36) 0 = ∂ GRADT g+ < f > T ρ−1Λf ∂ GRADT f #,

as well as to the residual dissipation inequality

(37) ΛM M ∗(Γ) + ΛE : E∗(Γ) + ΛAA∗(Γ) − T −2 QGRAD T ≥ 0,

where g := u−s(Λu)−1 ≡ u−T s is the free energy density. If the thermodynamicprocess is very near to equilibrium (cf. [28]) then the above residual inequalitypermits the direct application of Onsager-Casimir reciprocity relations. Someof the above Lagrange multipliers are explicitly given by:

ΛM = −ρ0

T −1∂ M

g− < f > Λf ∂ M

f #,

ΛE = −ρ0T −1∂ Eg− < f > Λf ∂ Ef #,

ΛA = −ρ0T −1∂ Ag− < f > Λf ∂ Af #,

whereas the others vanish

Λv = 0, Λ2 = 0, Λ3 = 0, Λ4 = 0, Λ = 0.

The details of the above given procedure are presented in the reference [28]where thermoplasticity of neutron irradiated steels was considered.

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3.2 Small magnetoelastic strains of isotropic plastically deformed

insulators and generalized normality

In order to illustrate the above derived constitutive and evolution equationswe accept in this subsection the following very simplifying assumptions for anisotropic body:

Assumption 3.2 Elastic and thermal strain as well as strain induced by re-versible and irreversible magnetization are small of the same order but plasticstrain itself is finite (cf. also [28])

Such an assumption corresponds to the so called piezomagnetism processeswhen magnetization is generated by straining processes (illustrated in [13, 26]).

Let us take into account that by its very nature the mechanical stress dis-appears when pure elastic strain vanishes and, similarly, the local magneticfield equals to zero if the reversible magnetization vanishes. Then, accordingto [2, 15] and the above relationship (5) it is reasonable to introduce magne-tostrictive quasi-plastic strain by means of

(38) Eµ = Lg.

Here L is the fourth rank tensor of magnetostriction constants symmetric onlyin indices of the first as well as the second pair whereas the notation: g ≡ −→γ ⊗−→γ stands for diadics of the unit vector of the magnetization vector (trg = 1).

For convenience the elastic strain tensor expressed in material convectivecoordinates, accepted in previous subsection, then reads:

Ee :=1

2FT p FT

µFT θ (FT

e Fe − 1)FθFµF p

≡ E − E p − FT p (Eθ + Eµ)F p.

(39)

As already discussed in detail in the first section, the constituents of the La-grangian strain tensor , namely, Ee, Eθ, Eµ as well as E p are incompatible.

With these facts taken into account and the above introduced assumption3.2 mechanical and magnetic constitutive equations are presented henceforth.First, mechanical part of the stress tensor must be linear in elastic strain (39).

In the case of elastic and thermal isotropy case we would have the relationship:

TK = D(E p)Ee ≡ (c11 + c2E p + c3E2 p) trEe + 2c4Ee +(40)

c5 (E pEe + EeE p) + c6(E2

pEe + EeE2

p),

which clearly exhibits a plastic strain induced anisotropy.Before formulating a magnetic constitutive equation, let us transform vec-

tors of magnetization and magnetic field into the corresponding second rank

14



skew-symmetric tensors. This is done by means of the product with material

convective Ricci third rank permutation tensor as follows:

(41) H ≡ E H = −HT , Ma ≡ E Ma = −MT a , a ∈ r, R.

In the above replacements

(42) Mr := M − MR

is the reversible magnetization vector while the skew-symmetric second ranktensors H, Mr and MR should replace corresponding vectors.

As for magnetic constitutive equation we first remark that we have acceptedHeavisisde-Lorentz form of Maxwell equations (cf. table on page 59. of [31]).Thus from the last of these equations i.e. (24) (i.e. CURL H = 0 ) we see that

the relationship between magnetic induction field vector B and the internal magnetic field vector H (opposing the local magnetic field vector) reads:

(43) H := B − J −1C M.

Due to the assumption 3.2 the magnetic constitutive equation for an isotropicferromagnetic must have the form:

(44) H = c7Mr + c8(E pMr + MrE p) + c9(E2 pMr + MrE2

p),

which is linear in reversible magnetization. The hereinabove constitutive ex-pression for H has been derived from (35) and (43) under assumption 3.1 bymeans of tensorial representations for the proper orthogonal group [49].10

Equation (40) is the generalized Hooke’s law accounting for plastic strain

induced mechanical anisotropy. It is noteworthy that the constitutive equationfor internal magnetic field predicts magnetic anisotropy induced by the samecause.

According to the assumption 3.2 the free energy function generating (40)and (44) must be quadratic in elastic strain and reversible magnetization i.e.:

g =1

2c1 i21 +

1

2c2 i22 +

1

2c3 i23 + c4 i4 + c5 i5 +

c6 i6 +1

2c7 i7 +

1

2c8 i8 +

1

2c9 i9,(45)

where the consecutive proper and mixed invariants appearing in the above scalarfunction must be introduced (cf. [49]):

i1 = trEe, i2 = trE pEe, i3 = trE2

pEe,

i4 = trE2e, i5 = trE p

E2e, i6 = trE2

pE2e,(46)

i7 = trM2r, i8 = trE pM2

r, i9 = trE2 pM2

r.

10The skew-symmetric tensors are favored instead of the corresponding vectors for conve-nience and more compact representation. Of course, an equivalent formulation using productsof vectors Mr and MR with symmetric second rank tensor Ep is also possible.

15



In the sequel inverse forms of (40) and (44) will be useful. They can be written

in the following way:

Ee = (γ 11 + γ 2E p + γ 3E2 p) trTK + 2γ 4 TK +(47)

γ 5 (E pTK + TK E p) + γ 6 (E2 pTK + TK E

2 p),

and

(48) Mr = γ 7H + γ 8(E pH + HE p) + γ 9(E2 pH + HE2

p).

The relationships between sets c1,...,c9 and γ 1,...,γ 9 can be found as fol-

lows. Let us multiply (40) as well as (47) by the tensors1

,E p and

E2

p findingtraces of both sides. If we introduce notations:

s1 = trTK , s2 = trE pTK , s3 = trE2 pTK ,

s4 = trT2, s5 = trE pT2K , s6 = trE2

pT2K ,(49)

s7 = trH2, s8 = trE pH2, s9 = trE2 pH2,

then such a procedure will produce relationships between c1,...,c6 and γ 1,...,γ 6.Of course, the same procedure applied to (44) as well as (48) would connectsets c7,...,c9 and γ 7,...,γ 9.

Similarly, the evolution equations for plastic strain rate and residual mag-netization rate are explicitly stated by the following formulae:

DtE p =< f >

d11 + d2Ee + d3(EeE p + E pEe) +

d4(EeE2 p + E2

pEe) + d5E p + d6E2

p +

d7(MrE p − E pMr) + d8(MrE2 p − E2

pMr) +

d9(MRE p − E pMR) + d10(MRE2 p − E2

pMR) +

d11(E pMrE2 p − E2

pMrE p) +

d12(E pMRE2 p − E2

pMRE p),(50)

DtMR = e1Mr + e2(MrE p + E pMr) + e3

MrE2 p+

E2 pMr

+ e4MR + e5(MRE p + E pMR) +

e6(MRE2 p + E2

pMR) + e7(E pEe − EeE p) +

e8(E2 pEe − EeE2

p) + e9(E2 pEeE p − E p

EeE2 p).(51)

16



It should be noted here that all the scalar coefficients in above relationships (40)-

(44) and (50)-(51) are functions of the principal invariants of the plastic straintensor E p. Of course, if plastic strain itself is small, then the correspondingcomplete linearization of constitutive and evolution equations is straightforwardwhich might be of interest especially if dynamic effects are considered i.e. waveequations of the linearized problem written (cf. [28]). Evolution equations thenwould reduce to Onsager-Casimir reciprocity relations.

The above evolution equations (50) and (51) become very much simplifiedif a generalized loading function Ω is assumed. Such a function would have theconsecutive orthogonality properties (cf. [30, 37])

(52) DtE p = DtΛ∂ Ω

∂ TK and DtMR = DtΛ

∂ Ω

∂ H.

where the material time rate of a scalar function Λ vanishes if the yield func-tions f as well as f # are either negative or zero (cf. (29)). These constitutiveequations (52) include magneto-mechanical reversible and irreversible interac-tions. To underline that the evolution equations are of quasi rate independenttype we use here DtΛ instead of Λ [40, 41]. A development of Ω into a powerseries whose terms are products of invariants (49) would give rise to differentmagnetomechanical irreversible couplings. Such a procedure has been appliedin [41] for thermomechanical processes. It is worthy of note that even in thesimplest case when loading function is quadratic function of H and T:

(53) Ω =1

2ω1 s21 +

1

2ω2 s4 +

1

2ω3 s7,

such a coupling may appear through DtΛ. This function leads by means of (52)into the consecutive two evolution equations

(54) DtE p = DtΛ (ω11 trTK + ω2 TK ),

(55) DtMR = DtΛ ω3H,

whose simplicity follows from the above very special loading scalar function Ω.They are oversimplified as will be seen from the following considerations.

3.3 Magnetoelasticity of cubic crystals by tensor representation

In order to show simplest magnetomechanical interaction let us consider in thissubsection magnetoelasticity of iron crystals. If we neglect the exchange energy,then the potential energy for a magnetic domain α ∈ 1, . . . , 6 inside a singlecrystal grain Γ reads (cf. [31], p. 378):

wαΓ =

1

2eE DeE + M 2s eE Bgα − µ0M s

−→H −→γ α+

1

4M 4s gαK1gα +

1

4M 6s gα(gαK 2gα)

(56)

17



with the notation: gα ≡ −→γ α⊗−→γ α (cf. (38)). Following [11] it has been assumed

here that strain eE and local magnetic field−→H are uniform for the considered

grain. Strictly speaking, all the terms on the right hand side of (56) except

µ0, M s and−→H should have subscript Γ to show grain dependence. For simplicity

all these subscripts are not written. In the above equation the first term onthe right hand side is the magnetoelastic energy wσ

Γ, the second term shows themagneto-strictive energy wσµ

Γ , magneto-static energy wmagΓ is the next term,

whereas the last two terms are responsible for the magneto-crystalline energywanΓ . If we subtract the magnetostrictive strain from magnetoelastic strain by

means of eαe ≡ eE − eαµ (cf. also (38)) and take into account that square of magnetostrictive strain is negligible (cf. also Appendix in [11]), then we get:

(57) wσµΓ + wmagΓ ≈ −tr(σαeαµ) − µ0M s−→H −→γ α.

For cubic crystals magnetocrystalline 4-tensor K1 and 6-tensor K 2 in thecrystallographic frame (CF) have the special forms [31]:

(K1)κλµν = K 1α

δακδαλδαµδαν ,

(K 2)αβκλµν = K 2δ1αδ1β δ2κδ2λδ3µδ3ν .

(58)

The base of (CF) is determined by the easy directions [100], [010] and [001]while unit vectors −→γ α, α ∈ 1, . . . , 6 are either parallel or antiparallel to this

base.For the sake of illustration the next figure shows the energy (56) for iron as

function of direction in the mentioned CF (cf. also [10]).An issue of high importance for polycrystals is concerned with volume frac-

tions of magnetisations inside magnetic sub-domains. Throughout a typicalcubic Λ-grain it is assumed along with [6, 10] that they fulfil the Boltzmanndistribution:

(59) f αΛ =exp(−AD)wα

Λβ exp(−AD)wβ

Λ

, α, β ∈ 1, . . . , 6.

It is important to note here that the first and last two terms in (56) are constantthroughout the Λ-grain. Therefore, only energies (57) changing from domainto domain actually appear in (59).

The coefficient AD was calibrated for iron by Daniel in his PhD-thesis [10] onthe basis of measurements by Webster performed in 1925. Such an identificationmeans that the coefficient AD depends either on experiment or material orboth11. Having found these volume fractions magnetisation vector inside Λ-

11As explained by Sommerfeld in [48] for small body immersed into a huge closed thermo-

18



Figure 2: Magneto-crystalline energy of iron

grain reads:

(60)−→M Λ :=

6

α=1f αΛ

−→M αΛ,

where constituents−→M αΛ are directed along the “easy” magnetic directions −→γ α,

namely, [100], [010], [001], [100], [010] and [001].In order to see how multiaxial stress influences magnetic susceptibility ten-

sor let us apply (59) and (60) to a grain of iron cubic crystal acted upon byarbitrarily oriented external magnetic field inside CF-plane [100] & [010]

−→H = H 0

cos(θH ) sin(θH ) 0

with θH ∈ [0, π/4] while H 0 ∈ [20, 2000]A/m.

To choose typical and representative stress histories let us remind that onlydeviatoric stress influences magnetostriction. Then the following two cases areworthwhile to be considered.

dynamic system the coefficient AD = 1/kT where k = 1.3806505 · 10−23[J/K ] is Boltzmannuniversal constant and T the absolute temperature. It is important to underline that thiscoefficient does not depend on process and energies. Such an approach was applied in thepaper [33] where field equations of Zorski’s statistical theory of dislocations [57] were closedby Boltzmann’s partition function in a very special case of 2D screw dislocations. Since amagnetic domain is much smaller than RVE a good preliminary choice would be to acceptSommerfeld’s suggestion. The essential difference b etween these two approaches is the ques-tion of universality of AD. Final judgement must be drawn from experiments.

19



Case 1. Let the stress tensor acting on the grain be

σ = σ0

cos2(θσ) sin(θσ)cos(θσ) 0sin(θσ)cos(θσ) sin2(θσ) 0

0 0 −1

with θσ ∈ [0, π/4] while stress magnitudes are varied in the interval σ0 ∈[0, 160]M P a. Thus its compressive component is along CF-[001] directionwhile the tension component is arbitrarily oriented inside plane [100] &[010].

Case 2. Let the stress tensor be situated in the same CF-plane as−→H i.e.

σ = σ0 cos(2θσ) sin(2θσ) 0

sin(2θσ) −cos(2θσ) 00 0 0

with θσ ∈ [0, π/4] and σ0 ∈ [0, 160]M P a. Since, in general, θσ = θH di-rections of external magnetic field vector and major principal stress differin general. In this case both principal stresses (tension and compression)are also arbitrarily oriented inside plane [100] & [010].

From the numerical results of the above two programs of magnetomechanical

histories with calculated values for−→M at this point we propose an identification

of stress and magnetic field dependent susceptibilities. Such a formula could beuseful for fast finding influence of stress on susceptibilities in each grain whenmagnetic constitutive equation for magnetization of a polycrystal is analyzed.

The fact that reversible magnetization depends on stress and magnetic fieldand it disappears with external magnetic field suggests tensor generators in thetensor representation formula (cf. [49]):

(61)−→M =

−→H α

χαΨα.

Thus, symmetric and skew-symmetric tensor generators are formed by products

of the tensors 1,σ and−→H ⊗

−→H . As a result we obtain

(62)−→M =

1

χ1 + χ2H 2 + χ3−→H σ

−→H

+ σ

χ4 + χ5H 2−→

H ≡ χ−→H .

Such a reduced representation is so chosen to have the smallest number of material “constants” χ1, . . . χ5. It has odd powers of magnetic field in order tomaintain magnetic symmetry requirement [31, 46]. Its linearity in stress is thesimplest approach taking into account magnetomechanical interaction. Whenstress disappears it fulfils symmetry requirement for cubic crystal.

20



Figure 3: Principal magnetic susceptibilities χ1/1000, 1000·χ2, χ3, 1000·χ4 andχ5 of mono-iron for diverse magnetic field H [A/m] and biaxial stress magnitudeσ0[M P a]. Out-of-plane case.

Input data for the numerical simulation are taken here for iron from [9, 11]as follows: λ[100] = 21 · 10−6 - magnetostriction constant for the easy direc-tion, µ0 = 4π · 10−7 - vacuum magnetic permeability, χ0 = 2000 - initialmagnetic susceptibility determined in Webster’s experiments (cf. [11]) andAD = 1.6 · 10−3[m3J −1] is Buiron–Daniel constant appearing in distribution(59). From these data the corresponding saturated magnetization equals to:

M s = (3χ0/µ0AD)1/2

[A/m]. Calibration results for both above cases are shownon the following two figures. Each (H max, σ0) point encompasses fitting for allχ-coefficients including all the points inside the 4-domain: H ∈ [0, H max], θH ∈[0, π/4], σ ∈ [−σ0, σ0], θσ ∈ [0, π/4]. Obviously, the proposed approximation(62) is satisfactory in domains where χ-coefficients are only slightly variable.

In order to present the results for χ-coefficients in a more explicit way letus introduce scaling coefficients κH ≡ H/H max and κσ ≡ σ/σ0. For simplicity,consider special case κH = κσ ≡ κ. Then dependence of correlation coefficient

21



Figure 4: Principal magnetic susceptibilities χ1/1000, 1000·χ2, χ3, 1000·χ4 and

χ5 of mono-iron for diverse magnetic field H [A/m] and biaxial stress magnitudeσ0[M P a]. In-plane case.

22



of magnetization calculated by (59) and (60) as well as by (62) on scaling coeffi-

cient κ can be determined. As an example let us take κ = 0.5 corresponding toH = 1000A/m and σ0 = 80M P a. For in-plane case calibrated constants readχ = 1726, −5.57 × 10−4, 6.97 × 10−4, 803, −1.21 × 10−3, η = 0.944 whereas forout-of-plane stress we have χ = 1661, −7.43 × 10−4, 1.12 × 10−4, 2250, −1.55 ×10−3, η = 0.896. The later correlation is not acceptable whereas the first ismuch better. From these two cases it may be concluded that the proposedtensor representation (62) is more suitable for in-plane case12. The specialstress-free case is easily observed from the figures 3 and 4.

At the end of this subsection let us remark that the formula (62) mightbe further improved introducing Boehler’s structural tensors for cubic crystals[4] at the expense of additional susceptibility coefficients13. Of course, such a

result is physically more justified due to correct material symmetries.Moreover in a polycrystal each grain orientation either is random or depends

on orientation function originating from texture. A comparison of simplifiedrepresentation (62) with the corresponding cubic crystal representation for fullrandomness is a worthwhile task.

It should be mentioned that Motogi and Maugin in [44] considered sub-domain distribution from convexity property of “stocked” energy g p (cf. (63))without introducing micro-distribution (59).

3.4 Approach by endochronic thermodynamics

1. Let us first briefly discuss purely mechanical inelastic irreversible behavior of steels given in [38]. The specific free energy of the considered body is takento be of the form

(63) g = gE (EE , T ) + g p (λ, T ) ,

where λ is the isotropic hardening parameter . Its time rate is given by

(64) Dtλ := TK : DtE p,

having the meaning of plastic power. Since the free energy is assumed in theform (63) we have the plastic part of dissipation

ℵ p = (1 − ρ∂ λg) Dtλ.

The total thermoplastic dissipation appearing in the second law of thermody-namics is denoted by ℵ, namely ℵ ≡ T

ρDts + div(q/T )

≥ 0, where ρ is the

12A further improvement of the results for the susceptibilities could be obtained if rotations

of domain magnetisations from−→H towards easy directions are calculated according to [7].

13Even the isotropic representation (62) predicts induced magnetic anisotropy i.e. different

orientations of −→H and

−→M (cf. [9] section 13.) unless stress disappears.

23



mass density, T is the absolute temperature, q is the heat flux vector and s is

the specific entropy.The plastic dissipation served Vakulenko to introduce his thermodynamic

time [53] by the next hereditary function

(65) ζ (t) :=

t 0

ψ

ℵ p(t)

dt.

The function ζ (t) is piecewise continuous and nondecreasing in the way thatDtζ (t) = 0 within elastic ranges and Dtζ (t) > 0 when plastic deformation takesplace. Splitting the whole time history into a sequence of infinitesimal segmentsVakulenko represented the plastic strain tensor as a functional of stress andstress rate history.

Moreover, in the paper [38] the accumulated plastic strain

(66) ε peq(ζ ) ≡

ζ 0

DtE p(ξ) dξ,

as the important inelastic history parameter, was included into the memorykernel, extending in such a way formerly mentioned Vakulenko’s arguments.Another important generalization of his model in [38] was extension of thefunction ψ to have the nonlinear power form:

(67) ψ(ℵ

p

) = (ℵ

p

)

a

.

The exponent a is of a great importance since it shows the speed of ageing. Forexample a < 1 may be named decelarated ageing whereas a > 1 would defineaccelarated ageing. By such a classification the Vakulenko’s value a = 1 mightbe termed as steady ageing.

Now, according to Vakulenko’s postulate we have:

(68) E p(ζ ) =

ζ 0

Ψ

ζ − ξ, TK (ξ), DξTK (ξ), ε peq(ξ)

dξ.

Of course, this integral equation is adopted to our case of finite plastic strainsand absence of plastic rotation. Differentiation of (68) with respect to thethermodynamic time gives

∂ ζ E p =Ψ

0, TK (ζ ), Dζ TK (ζ ), ε peq(ζ )

+

ζ 0

∂ ζ Ψ

ζ − ξ, TK (ξ), DξTK (ξ), ε peq(ξ)

dξ.(69)

24



Further analysis of the above integral equation is given in the next subsection.

2. Let us apply now the above explained concept to evolution of irreversiblemagnetization. Again we have non-steady ageing speed defined by the exponenta by means of:

(70) Dtζ = (ℵPM )a ≡

HDt M R + tr(TK DtE p)

a,

but now irreversible power induced by magnetisation must be taken into ac-count. It is included into the magnetoplastic dissipation ℵPM . Suppose nowthat magnetomechanical interaction is only through equivalent plastic strainhistory. Then the magnetic evolution equation in its integral form may betaken as

(71) M R(ζ ) :=

ζ 0

Ψ(ε peq(z), ζ − z, H (z)) dz,

where the corresponding endochronic memory is characterized by the thermo-dynamic time (70). Choosing a special form of the integral kernel as follows14

(72) Ψ = H (z) ω(ε peq) exp −β (ζ − z)

we would arrive at the following simple explicit evolution equation for residualmagnetization vector:

(73) Dζ M R = ω H − β M R.

However, in this equation derivative is taken not with respect to real but tothermodynamic time. In order to transform it to real time let us first introduceirreversible magnetic power by means of

(74) Dtλµ := HDt M R,

following the same notation in (64). Now, when we replace (74) into (70) andmultiply this by magnetic field vector and time derivative of thermodynamictime we get a nonlinear algebraic equation:

(75) Dtλµ

ω| H |2 − H M R

a1−a

− Dtλµ := Dtλ.

This equation explicitly characterizes magnetoplastic interaction. Its validityshould be checked by experiments where simultaneously stress, plastic strain,magnetic field and residual magnetization are measured. The two interestingspecial cases may be drawn from this equation:

14Such an exponential kernel is typical for endochronic theories (cf. [54]).

25



If plastic power is approximately equal to zero, then we would have a ther-

moelastic irreversible magnetization . Since Dtλ ≈ 0 the equation (75)gives a simplified time rate of the thermodynamic time:

(76) Dtζ = (Dtλµ)a =

ω| H |2 − H M R

a1−a

.

Another special case of interest would be choice a = 1 which might be calledVakulenko’s coupled magneto-viscoplasticity . For such a choice time rateof ζ reads:

(77) Dtζ = Dtλµ + Dtλ =Dtλ

1 − ω| H |2 + H M R.

In both cases evolution equation for residual magnetization in real time domainhas the form:

(78) Dt M R =

ω H − β M R

Dtζ.

Suppose for simplicity that constitutive equation (62) holds not only for agrain but for RVE as well. Then M r = χ H and using (43) we arrive at theintegral evolution equation connecting magnetic induction and magnetic fieldvectors:

B(ζ ) = H (ζ ) + J (ζ )−1

C(ζ )χ(ζ ) H (ζ )+ζ 0

Ψ(ε peq(z), ζ − z, H (z)) dz

,(79)

where the explicit form of the right Cauchy-Green total deformation tensor15

C = 1 + 2E is found from the relationship (39). Here magnetomechanical in-teraction appears through total as well as plastic strain history and fact thatsusceptibilities depend on stress. Obviously, the oversimplified equation (55)might hold only in the case of negligible β. We believe that the above integralequation could be used for some nondestructive experimental check of order of magnitude of magnetomechanical interactions at low cycle fatigue or at someother experiments designed to establish characteristic points of inelastic behav-ior of steels or some other ferromagnetic materials.

15Terminology is taken from [52] Sec. 24.

26



3.5 Low-cycle fatigue of ferromagnetics

The constitutive and evolution equations described in previous subsectionsmight be used to describe piezomagnetic behavior induced by low-cycle fatigueof ferromagnetics.

Such a process has been investigated in the paper [13]. A cylindrical speci-men of AISI 1018 was uniaxially treated by push-pull tests on MTS-810 servo-hydraulic testing machine such that total strain was periodic and triangularlyshaped E ∈ 0, 0.009 with cycle duration of 2 s. Magnetic induction dueto piezomagnetic effect was also almost periodic with very slight changes of periodicity with increase of relative number of cycles N/N f and cumulation of phase delay with respect to strain with growth of accumulated plastic strain.Maxima and minima of total Lagrangean strain E are displaced with respectto minima and maxima of the magnetic induction vector16. According to (66)we calculate accumulated plastic strain as a function of time by means of

(66’) peq(t) :=

t0

DtE p(τ ) dτ.

Now, if uniaxial components of tensors E, B, M r, M R are denoted by meansof E 11, B1, M r1, M R1 then the following memory-type equation emerging from(55) and (71)

(80) B1

(t) := t

0J ( p

eq, t − τ ) D

τ λ(τ ) H

1(τ ) dτ,

could cover the delay between measured functions E 11(t) and B1(t). Timedifferentiation of the above relationship gives rise to the expression:

DtB1(t) := J ( peq, 0) Dtλ(t)H 1(t)+ t0

∂

∂tJ ( peq, t − τ ) Dτ λ(τ ) H 1(τ ) dτ.

(81)

In the above integro-differential equation the second term on the right hand sideis responsible for the above mentioned change of time delay and the deflection

of pure periodicity of B1(t). Therefore, it is much smaller than the first part.On the other hand, if the constitutive equation

−→B = µ0

(1 + χ)

−→H +

−→M R

(by

means of (62)) is used, then we would have

DtB1 = µ0DtM R1 + µ0Dt

(1 + χ11)H 1

,

DtE 11 = DtE e11 + DtE µ11 + DtE p11.(82)

16Here for convenience again magnetic induction is represented by the vector−→B .

27



Now, the relationships (77) and (64) in our case lead to:

(83) DtM R1 =

ωH 1 − βM R1 σ11DtE p11

1 − ωH 21 + H 1M R1.

The above three equations show a clear magnetomechanical interaction. Theymay serve for identification of material functions from LCF uniaxial tension

experiments (like those in [13]). Simultaneous zeros of DtE p and Dt−→M R (en-

suing from (54) and (55)) are consequence of the model where it is assumedthat irreversible magnetisation and plastic strain are triggered at the same timeinstant.

Instead of using (82) and (83), it is possible to find a more explicit en-dochronic kernel in (81). Let us remind that the Langevin function is suggestedin many references as the best approximant for anhysteretic curve17 (cf. for in-stance [43]). Then using the data in (18.137) and figure 22.1 from [9] for suchcubic crystals18 it is possible to depict the consecutive figure by translatingthe anhysteretic curve by coercive magnetic field H c either to the right or tothe left depending on sign of time rate of magnetic field. Dropping indices forsimplicity this may be represented by the following kernel

Ψ(ζ, z) = M 0

δ

H (z) − H (ζ ) + H c(λ)sgn(DtH )

−

δ

H (z) − H (ζ )

L

H (z)

DzH,

(84)

−0.06 −0.04 −0.02 0 0.02 0.04 0.06

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

H [T]

M [T]

Hc

MR

0

Anhysteretic curve

Figure 5: Soft ferromagnetic steel behavior approximated by Langevin functionaccording to [9]

17Let us recall that this function has the form L(H ) = coth(H ) − 1/H.18In the case of a soft ferromagnetic steel this author suggests the following data: (M )|H=0 =

0.832M sat, (dM/dH )|max = 0.6M sat/H c where H c = 0.0063T is the coercive magnetic fieldand M sat = 2.15T is the value of saturated magnetization value.

28



where sgn(x) = 1 for x > 0 and sgn(x) = −1 for x < 0. If this kernel is

inserted into the integral equation (71) then it gives rise to the following explicitexpression for residual magnetization:

M R = M − M r =

M 0 L

H − H c sgn(DtH )

− M 0 L(H ).(85)

It is worthy of note that herein the magneto-mechanical interaction is takeninto account by dependence of coercive field on plastic power λ whose time rateis given by (64).

Turning again to the paper [13] it may be concluded that the dependenceH c(λ) and (85) permit taking into account complex disturbances of shape from

simple periodicity of E 11(t) towards more complicated shape of B1(t) as well astheir relative delays of minima and maxima. It may be thus concluded that suchan equation could be fruitful for description of magneto-viscoplastic phenomenaat low cycle fatigue.

4 Effective field method for ferromagnetic polycrys-

tals

From this point we will assume that in the relationship (39) all the strains aresmall such that an approximate additivity holds. Then Eulerian strains fulfilthe following relation:

(86) e = ee + eθ + eµ + e p ≡ ee + eωp,

with: 2eα = 1 − F−T α F−1

α , α ∈ e,θ,µ,p. Here the term eωp is the inelastic“eigen-strain” (with thermal, magnetostrictive and plastic parts) representinga source of internal stresses (cf. [18]). Suppose that a RVE is composed of many grains which may be modeled by randomly oriented ellipsoids. If sucha RVE has homogeneous elastic strain, magnetization and temperature then itis customary for the effective field method (cf. [18]) to consider such a grainas an inclusion implanted (by “eigen-strains”) into a hypothetical matrix withaverage properties over grain orientations (with notation 1

N g Γ AΓ ≡ AΓ):

(87) D0 = DΓ, L0 = LΓ, α0 = αΓ,

Two main hypotheses of effective field method (according to [23, 18]) are:

Hypothesis 4.1 Inside each ellipsoidal inclusion strain is homogeneous.

Hypothesis 4.2 Ergodic property holds i.e. properties of inclusions are sta-tistically independent of their spatial distribution.

29



The first hypothesis in [27] is named “step-constant approximation”. If these

two hypotheses are fulfilled and the RVE is acted upon by some “external ”stress σ0, then performing the procedure applied in [24, 23] the stress andstrain inside a Γ-grain become:

(88)σΓ = σ0 +

∆ S0 ∗

[[D−1∆ ]]σ∆ + [[eωp∆ ]]

δ∆,

eΓ = e0 −

∆K0 ∗

[[D∆]]ee∆ −D0[[eωp∆ ]]

δ∆,

where (A ∗ a)(x) := A(x − x)a(x)dx, A ∈ K0, S0, [[a∆]] ≡ a∆ − a0 is

the jump of a across ∆-grain boundary and δ∆ is the characteristic function of ∆-grain being unity when position vector points to ∆-grain and zero otherwise.

The corresponding characteristic function for whole RVE is obtained by sum-mation of characteristic functions of all the grains belonging to the consideredRVE. The kernel S0 was determined in [18]:

(89) S0(x) = D0K0(x)D0 − D0δ(x)

by means of the kernel (in Kunin’s notation [23]):

(90) K0 = −def G0def

built by the Green function G0 corresponding to stiffness D0 (cf. (87)) anddef −→a ≡ sym( ⊗ −→a ).

For simplicity, let us concentrate our attention to the special case of magne-toelastic strain leaving out plastic and thermal strains. Then effective stiffnessand effective magnetostriction tensor are defined by means of

(91) σΓ = Deff eeΓ, eµΓ = Leff gΓ,

Here volume averages give macroscopic elastic strain, macroscopic stress, macro-scopic magnetostrictive strain and magnetization dyadic:

ee = eeΛ , T = σΛ , eµ = eµΛ , g = gΛ ,

respectively.

Even in this linear case without thermal and plastic strains for ellipsoidsrandomly oriented with crystallographic frames inside them incoincident withtheir semi-axes solution of the set of coupled integral equations (88) towardshomogenization formulae (91) is tremendous. For this reason some reasonablesimplifications are inevitable.

In homogenization theories for composites with particulate inclusions thereare two distinct self-consistent approaches:

30



• effective medium approach where it is assumed that each inclusion behaves

as isolated and immersed into a medium having effective constants Deff

and

• effective field approach with an assumption that again each inclusion be-haves approximately as isolated and situated into the matrix with elas-ticity constants DM while influence of neighboring inclusions is taken intoaccount by means of the effective strain field eeff acting on the consideredinclusion [27].

In this paper the second approach is employed. The mentioned effective strainin the case of pure elastic strain covers in the second of (88) all the terms inthe sum with ∆ = Γ under the assumption that correlation field induced by all

the other inclusions on Γ-grain is of the same shape as Γ-grain itself but withlarger dimensions. This assumption is simply illustrated by the figure 6. Under

(RVE)

e0

e, A

, Aeeff F

Figure 6: Effective self consistent field homogenization

such assumption the authors in [19] found (I is unit 4-tensor):

(92) eeff (x) = (I− pΓAΦ[[DΓ]]MΓ)−1 e0,

withMΓ = (I+A(xΓ)[[DΓ]])−1,A(x) =

K0(x−x)dx andAΦ =

K0(x)Φ(x)dx.

The function Φ(x) describes correlation distribution inside RVE. Having foundeeff the average strain equals to

(93) eΓ = MΓeeff .

Substituting this expression into (88) leads to the effective stiffness 4-tensor:

(94) Deff = D0 +Γ

pΓ[[DΓ]]I+

A(xΓ) − pΓAΦ

[[DΓ]]

−1.

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In the special case when all the inclusions have parallel semi-axes coinciding

with their crystallographic frames we have A(xΓ) = AΦ (∀Γ ∈ 1, N g) andthe above relationship takes the form of Mori-Tanaka effective stiffness: Deff =

D0 +

Γ pΓ[[DΓ]] (I + (1 − pΓ)A[[DΓ]])−1 .Suppose now that throughout a RVE magnetostrictive strains are balanced

in the way that average stress induced magnetostrictive grain strains equals tozero. Then by making use of the ergodic hypothesis 4.2 we obtain

(95) LT eff σ0 − L

T 0 σ0 − LT Γ σΓ = 0,

where (LT )cdab = (L)abcd. Due to linearity of (88) we may introduce the stressconcentration 4-tensor by means of the substitution σΓ = NσΓ σ0. Knowing thisstress concentration tensor we find the effective magnetostriction 4-tensor by:

(96) LT eff = L

T 0 + [[LT Γ ]]Nσ

Γ .

This concentration tensor is found from the reduced version of first of (88) wheninelastic terms are dropped, i.e.

(97) NσΓ = I+

∆

S0 ∗ [[D−1∆ ]]Nσ∆.

Omitting details of the derivation we just quote result for the tensor LT eff derivedin the same way as effective thermal expansion tensor in [24]:

(98) LT

eff = L

T

0+ B

Γ−1 ∆ B

∆[[LT

∆]]

with B∆ =I + D0

I − A(x∆)D0

[[D−1∆ ]]

−1. Let us finally quote the effective

thermal expansion tensor derived by Levin in [24]:

(99) αeff = α0 + BΓ−1∆

B∆ [[α∆]] .

In this way we have completed all the necessary effective constitutive ten-sors entering Hooke’s law for RVE based on microstructural thermo-magneto-mechanical properties of individual grains. It is noteworthy that effective mag-

netostriction as well as effective thermal expansion tensors are derived frompure elasticity consideration. According to Taylor’s assumption plastic strainis assumed to be homogeneous throughout RVE being equal for all the grains.19

19Let us note that the effective field approach to effective susceptibilities by means of grainbased constitutive equation is more complicated than the above analysis due to essentialnonlinearity of (62). It seems that a variational approach like in [55] is more suitable. Anyway,

if we suppose that−→H Γ =

−→H Γ, then the Wiener upper bound dependent on RVE-average of

stress could be obtained (cf. also [11]).

32



5 Some concluding comments

The subject has been treated by tensor representation applying either non-associativity of flow rule with extended thermodynamics or generalized normal-ity which includes orthogonality of residual magnetization rate on generalizedloading surface which includes mechanical as well as magnetic state variables.While plastic strain is finite thermoelastomagnetostrictive strain is assumedto be small. Small magneto-elasto-viscoplastic strains are then considered indetail in order to analyze magnetomechanical interaction at low-cycle fatigue.Furthermore, endochronic thermodynamics with Vakulenko’s thermodynamictime made possible an account to (experimentally observed) time delay betweenstress and magnetic field histories. Such a result could be useful in inelastic

testing with magnetic fields either induced or applied.Geometrical approach based on early papers in the field of continuum theory

of dislocations leads to the essential difference between micro and macro-spinhaving as the origin constrained micro-rotations of grains inside a representativevolume element. Here an Eshelbian approach is applied assuming that quasi-plastic (thermomagnetoplastic) strain is unconstrained whereas elastic strain isconstrained. Since a RVE, having volume of an infinitesimal volume element,cannot be disintegrated any more, micro-spin does not follow from plastic micro-stretching. In previous papers of the author (reviewed shortly in [41]) treatingpurely thermomechanical strain histories of viscoplastic polycrystals such anidea has been proved to be very successful. Namely, purely elastic micro-strains

have been assumed to be covered by self consistent method (effective mediumor effective field approach by Levin) whereas for plastic stretching as well asresidual magnetization rate quasi rate independent incremental macro-evolutionequations are postulated. The rate dependence takes place by means of stressrate dependent value of the initial yield stress. The macroscopic magneto-inelastic evolution equations obey the Vakulenko’s concept of thermodynamictime. The macro-evolution equation for plastic spin of RVE is an outcome of the corresponding evolution equation for plastic stretching. The same does nothold true for plastic micro-spin.

This paper has been dealt with viscoplasticity of ferromagnetic materials.Evolution equations have been derived either from inelastic materials of differ-

ential type or from loading function generalized normality. In both cases tensorrepresentation is applied to such a set of evolution equations. Restrictions tothe set of field equations are established by means of the extended irreversiblethermodynamics (version which follows exposition in [8]). Small magnetoelas-tic strains of isotropic insulators are considered in detail in two special casesof finite as well as small plastic strain. As one example low-cycle fatigue of ferromagnetics is considered with special account to time delay between stressand magnetic field histories. To describe such an experimental evidence an

33



integro-differential equation is proposed whose equivalent plastic strain depen-

dent kernel covers the observed delay.Concluding this section it is inevitable to compare the foregoing results with

existing achievements in the field. The major contributions to viscoplasticityof ferromagnetic materials have been given by Maugin and his collaborators in[30, 31]. The principal assumptions accepted in this section are closer to thescope of the first of these references where

small strain case together with absence of exchange forces and gyromag-netic effects has been assumed;

the accent on hysteresis effects has been given and

evolution equations have been derived by normality of plastic strain rateand residual magnetization rate onto a loading surface.

On the other hand we presented here the following results:

in the case of finite plastic strains magnetic anisotropy induced by plasticstrain is predicted by (51) where development of residual magnetizationby mechanical terms is also evident;

the influence of magnetization on plastic strain rate is obtained even inthe case of isotropic ferromagnetic materials;

the extended thermodynamics procedure allows for more general historyeffects with inhomogeneities of magnetization taken into account;

the obtained relationships with couplings allow for magnetic measure-ments of inelastic phenomena but the measurements will show their orderof magnitude and practical measurability of these phenomena;

in general, the developed theory is of non-associate type for plastic strainrate and residual magnetization rate are not perpendicular to the yieldsurface;

albeit a generalized normality is much simpler with smaller number of

material constants, a careful examination of the experiments on piezo-magnetism and magnetostriction processes would give the final judgementwhich theory should be applied;

endochronic thermodynamical approach is less general than the approachby extended thermodynamics but it is much more suitable for explicitdescription and calibration of inelastic magneto-mechanical experimentslike low cycle fatigue stress-strain-induction histories.

34



Acknowledgment

The author is grateful to Professors Gerard Maugin and Vukota Babovic forvaluable discussions concerning some aspects of electromagnetism.

Reviewer’s criticism has helped very much to clarify and organize the textof this paper.

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ZorskiAMS Micunovic Revised for Reviewer