Universitat Karlsruhe (TH)

Institut fur Baustatik

A constitutive model for

magnetostrictive and piezoelectric materials

K. Linnemann, S. Klinkel, W. Wagner

Mitteilung 2(2008)

BAUSTATIK

Universitat Karlsruhe (TH)

Institut fur Baustatik

A constitutive model for

magnetostrictive and piezoelectric materials

K. Linnemann, S. Klinkel, W. Wagner

Mitteilung 2(2008)

BAUSTATIK

c

Prof. Dr.Ing. W. Wagner Telefon: (0721) 6082280Institut fur Baustatik Telefax: (0721) 6086015Universitat Karlsruhe Email: bs@.uni-karlsruhe.dePostfach 6980 Internet: http://www.bs.uni-karlsruhe.de76128 Karlsruhe

A constitutive model for magnetostrictive andpiezoelectric materials

K. Linnemanna, S. Klinkelb, W. WagnerbaFraunhofer Inst. fur Kurzzeitdynamik, Ernst-Mach-Instiut,

Eckerstrae 4, 79104 FreiburgbInstitut fur Baustatik, Universitat Karlsruhe (TH),

Kaiserstrae 12, 76131 Karlsruhe

Contents

1 Introduction 3

2 Phenomenological behavior 52.1 Magnetostrictive materials . . . . . . . . . . . . . . . . . . . . . 62.2 Ferroelectric ceramics . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Governing equations 93.1 Magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Thermodynamic framework 124.1 Integration algorithm . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Hardening function . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Algorithmic consistent tangent moduli 15

6 Ferrimagnetic hysteresis effects 16

7 Constitutive modelling of ferroelectric ceramics 17

1

8 Variational formulation and finite element approximation 178.1 Element formulation . . . . . . . . . . . . . . . . . . . . . . . . 18

9 Numerical simulations 209.1 Magnetostrictive hysteresis . . . . . . . . . . . . . . . . . . . . . 209.2 Ferroelectric hysteresis . . . . . . . . . . . . . . . . . . . . . . . 229.3 Amplified magnetostrictive actuator . . . . . . . . . . . . . . . . 239.4 Piezoelectric hollow cylinder . . . . . . . . . . . . . . . . . . . . 279.5 Piezoelectric telescope actuator . . . . . . . . . . . . . . . . . . 32

10 Conclusion 33

A Vector notation and linear material constants 35

B Scalar valued conjugated variable to the internal variable 36

C Derivatives for the integration algorithm 36

D Derivatives for the algorithmic consistent tangent moduli 37

Abstract This paper is concerned with a macroscopic nonlinear constitu-tive law for magnetostrictive alloys and ferroelectric ceramics. It accounts forthe hysteresis effects which occur for the considered class of materials. Theuniaxial model is thermodynamically motivated and based on the definitionof a specific free energy function and a switching criterion. Furthermore, anadditive split of the strains and the magnetic or electric field strength into areversible and an irreversible part is suggested. Analog to plasticity, the irre-versible quantities serve as internal variables. A one-to-one-relation betweenthe two internal variables provides conservation of volume for the irreversiblestrains. The material model is able to approximate the ferromagnetic or ferro-electric hysteresis curves and the related butterfly hysteresis curves. Further-more, an extended approach for ferrimagnetic behavior which occurs in mag-netostrictive materials is presented. A main aspect of the constitutive modelis its numerical treatment. The finite element method is employed to solve thecoupled field problem. Here the usage of the irreversible field strength permitsthe application of algorithms of computational inelasticity. The algorithmicconsistent tangent moduli are developed in closed form. Hence, quadraticconvergence in the iterative solution scheme of governing balance equations isobtained.

Key words Nonlinear constitutive law, magnetostrictive alloys, ferro-electric ceramics, ferromagnetic hysteresis, smart material, computationaltreatment, finite element method

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1 Introduction

Magnetostrictive alloys and ferroelectric ceramics are smart materials. Theyhave a wide range of application for actuation and sensing, see e. g. Smith(2005). Magnetostrictive materials show an inherent coupling between mag-netic field and deformation. This effect bears a resemblance to piezoelectric-ity which describes a coupling between electric field and deformation. Bothmaterials show similar nonlinear behavior. The purpose of this paper is thedevelopment of a constitutive model which accounts for the nonlinear behaviorand hysteresis effects of magnetostrictive alloys and ferroelectric ceramics.

Microscopically motivated models approximate the switching processes foreach single crystallite. The process is modeled by an energy criterion. Themacroscopic behavior is obtained by averaging over a large number of unioncells. Microscopically motivated models for ferroelectric ceramics are presentedin e. g. Chen and Lynch (2001), Hwang et al. (1995, 1998) and Huber and Fleck(2001). Approaches for magnetostrictive materials are often based on themodel of Jiles and Atherton (1986) which was primarily developed for ferro-magnetic hysteresis without magnetoelastic coupling. The formulation ap-proximates the bending and movement of the domain walls during the mag-netization. Extended models for magnetostrictive material are found in e. g.Sablik and Jiles (1993), Jiles (1995), Calkins et al. (2000), Dapino et al. (2000b),Dapino et al. (2000a), and Dapino et al. (2002). With the consideration ofswitching for each crystallite microscopically motivated models generally leadto a large number of internal variables which increases the numerical effort.

Macroscopical constitutive models are based on a phenomenological de-scription of the material behavior. This reduces the amount of internal vari-ables. A widespread approach is the model of Preisach (1935) which wasoriginally developed to describe the magnetization of ferromagnetics. Thechoice of parameters allows the simulation of a wide range of hysteresis curves.Formulations for magnetostrictive alloys are proposed in Adly et al. (1991)and Tan and Baras (2004). Applications of the Preisach model for piezoelec-tric hysteresis are published in Hwang et al. (1995), Pasco and Berry (2004),Yu et al. (2002a,b) and Butz et al. (2005, 2008). The Preisach model is purelyphenomenological and not thermodynamically consistent.

Another class of constitutive models are formulated in accordance withthe principles of thermodynamics. The material behavior is determined withthe definition of a specific free energy function. For piezoelectric ceramicsthe strains, the polarization, and the temperature often serve as independentvariables. For magnetostrictives the magnetization is used instead of the polar-ization. Here, a common approach are higher order energy functions, see e. g.Carman and Mitrovic (1995), Wan et al. (2003), Zheng and Liu (2005), andZheng and Sun (2007). Alternatively a definition of the free energy functionin sections is used, see Wan et al. (2003) and Smith et al. (2003). The indepen-dent variables are splitted in a reversible and an irreversible part for the approx-

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imation of the hysteresis behavior. The irreversible quantities serve as internalvariables and represent the polarization or magnetization state. The develop-ment of the internal variables is determined with the definition of a switchingcriterion. The model of Fang et al. (2004) for magnetostrictive materials isbased on these assumptions. More often, thermodynamically motivated mod-els are applied for piezoelectric ceramics. The approaches of Kamlah (2001),Kamlah and Bohle (2001) and Elhadrouz et al. (2005) use multiple switchingcriterions to specify begin and end of the irreversible behavior. The multiaxialmodels of Landis (2002) and McMeeking and Landis (2002) get along with onlyone switching criterion. The domain processes are restricted by a hardeningfunctions. Schroder and Gross (2004) and Schroder and Romanowski (2005)present a co-ordinate invariant formulation on this assumption. The modelsof McMeeking and Landis (2002), Kamlah (2001), Schroder and Romanowski(2005) use a one to one relation between irreversible strains and polarizationto ensure that domain switching induces irreversible strains. This assump-tion is sufficient for electrical loading but prevents the simulation of ferroelas-tic behavior or mechanical depolarization. The circumvent this disadvantageKamlah (2001) decomposes the irreversible strains in a part for ferroelectricand one for ferroelastic switching processes. A more general approach is pub-lished in Landis (2002) and Klinkel (2006a,b). Here a coupled switching criteri-ons and hardening functions are used to control the change of the irreversiblestrains. Mehling et al. (2007) introduce an orientation distribution functionas additional internal variable to approximate coupled electromechanical load-ings.

In the present paper a thermodynamic motivated constitutive model isdeveloped which accounts for hysteresis effects in ferroelectric ceramics andmagnetostrictive alloys. The uniaxial model is embedded in in a three di-mensional formulation. This benefits an efficient numerical treatment of themodel but prevents a reorientation of magnetization or polarization duringthe simulation. The electric respectively the magnetic field strength is splittedadditively. The irreversible field strength serves with the irreversible strainsas internal variable. The main aspects of the model may by summarized asfollows:

A thermodynamically consistent constitutive model is presented. Theformulation is based on the definition of a free energy function and aswitching criterion. A polynomial of second order is used as free energyfunction. The switching criterion controls domain switching. The centerof the switching surface moves in the sense of kinematic hardening inplasticity.

The strains, the magnetic and the electric field strength respectively aredecomposed additively in a reversible and an irreversible part. The splitof the field strength is an alternative approach to the decomposition ofthe magnetization or polarization. Its application is motivated by the

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numerical treatment of the constitutive model with the finite elementmethod. Here the field strength is described with a scalar potentialwhich serves as nodal degrees of freedom. So no change of variables isnecessary. Furthermore, one may draw on algorithms of computationalinelasticity.

A one to one relation between the irreversible quantities is utilized tomaintain volume conservation during the irreversible processes. A hard-ening function restricts the growth of the internal variable near satura-tion.

The model is able to reproduce the ferroelectric and butterfly hysteresisof ferroelectric ceramics. Furthermore, the typical hysteresis of magne-tostrictive alloys can be simulated. The occurring ferrimagnetic hystere-sis behavior of these materials is approximated by an expanded approachwhich is based on a transformation of the independent variables in a lo-cal description. Ferroelastic switching and mechanical depolarization arenot considered within this work.

For the numerical treatment of the problem the finite element method isemployed. The constitutive model is implemented in a hexahedral ele-ment. Therefore the evolution equation is integrated by an implicit Eu-ler backward algorithm which leads to a local iteration. The algorithmicconsistent tangent moduli are formulated in closed form. Accordingly,quadratic convergence in the iterative Newton-Raphson solution schemeis guaranteed.

The outline of the paper is as follows: Section 2 deals with the phenomeno-logical behavior of magnetostrictive and piezoelectric materials. In Section 3the governing equations of the coupled field problems are introduced. Thethermodynamic framework of the constitutive model is presented in Section4. In Section 5 the algorithmic consistent tangent moduli are developed. Anexpanded model for ferrimagnetic behavior is presented in Section 6. Section 7deals with the modeling of ferroelectric ceramics. In Section 8 the variationalformulation and the finite element approximation are presented. The numeri-cal examples in Section 9 show the capabilities and main characteristics of theproposed constitutive model.

2 Phenomenological behavior

This section summarizes the behavior of the considered smart materials. Atfirst we account for the origin and phenomenology of magnetostriction. Forextended treatises on magnetism of solids, the books of Bozorth (1964) andEngdahl (2000) are recommended. The second part of the section is concernedwith piezoelectric coupling, in particular with ferroelectric ceramics. It briefly

5

deals with the differences to the magnetostrictive model. Piezoelectricity iswell documented in literature, see Jaffe et al. (1971) and Ikeda (1990). A goodsurvey of the basic phenomenons of ferroelectric ceramics and their nonlineareffects is given in e.g. Kamlah (2001).

2.1 Magnetostrictive materials

The term magnetostriction is applied for several magneto-elastic coupling phe-nomena. The most utilized effect is the Joule magnetostriction which denotesa deformation of a specimen due to an applied magnetic field H . The recipro-cal Villari effect is characterized by a change of the magnetization M inducedby a mechanical deformation. Both phenomena occur in all ferro-, ferri, andantiferromagnetic materials. The magnetic properties of these materials orig-inate from magnetic moments m which results from the spins of electrons inuncomplete occupied inner orbitals of the atom. Due to spin orbit coupling,which is a purely quantum mechanical interaction, the magnetic moments areclosely connected with the shape of the electron hull e. The electrical nega-tive electron hull effects Coulomb forces to the neighboring atoms in the lattice.Regarding the complex shape of e these forces are not isotropic, which is im-portant for magnetostrictive coupling. This correlation is described by meansof a sample crystallite depicted in Fig. 1. Above the Curie temperature Tc themagnetic moments are disordered, see Fig. 1a). Cooling the material downbelow Tc, the exchange interaction causes a parallel ordering of the atoms inthe lattice, see Fig. 1b). Applying an external magnetic field the magnetic mo-

ments switch in the direction of H . This process is associated with the rotationof the electron hulls e. Along with the rotation of e the Coulomb forcesbetween the atoms change and the gaps in the lattice shift. Consequently, adeformation of the specimen is observed, see Fig. 1c). The magnetostrictivestrains are denoted by and . The volume conservation of the deformationinvolves = 12. Depending on the electron configuration some materialsshow a positive magnetostriction with > 0. For materials with negativemagnetostriction it holds < 0.

Figure 1: Illustration of the Joule magnetostriction

To illustrate the origin of the Villari Effect a crystallite with positive mag-

6

netostriction below Tc is considered, see Fig. 2a). The sample is deformed bya mechanical stress S parallel to m. The deformation shifts the inter-atomicdistances in the lattice. This change makes sure that another direction of themagnetic moments is energetically convenient. So the magnetic moments ro-tate as depicted in Fig. 2b). Since no direction of the magnetic moments isspecified by S, the orientations of Fig. 2b) and c) have the same probability.This results generally in a reduction of the magnetization for polycrystallinematerials.

Figure 2: Illustration of the Villari effect

Parallel aligned magnetic moments are energetically not favorable on meso-scopic scale. Hence, the crystallites are divided in substructures, the so-calleddomains. The magnetic moments in each domain are aligned parallel, which iscaused by the exchange interaction. The neighboring domains have an oppositemagnetization direction to minimize the magnetic energy. They are separatedby domain walls. The structure of a polycrystalline material is shown in Fig. 3.

crystallite domain wall

Figure 3: Domain structure of a magnetostrictive alloy

The nonlinear behavior of magnetostrictive materials is strongly connectedwith the domain structure. If a magnetic field is applied to a specimen, themagnetic moments along the domain walls begin to rotate in the direction ofH . The domain walls begin to move. Due to discontinuities in the latticethe movements of the domain walls are irreversible processes. Hence, for anoscillating magnetic field hysteresis curves are observed. In the ferromagnetichysteresis curve the magnetic flux density B is plottet versus the field strengthH . The curve of Fig. 4a) is exemplary plotted for the X1 direction. The

7

ordinate of the so called butterfly hysteresis in Fig. 4b) is the strain componentE11. The domain states are sketched for both figures. The characteristic valuesof the curves are the coercive field strength Hc and the remanent values Bremand Erem.

Figure 4: Ferromagnetic and butterfly hysteresis curves in magnetostrictivematerials

Commercial available magnetostrictive alloys like Terfenol-D show a ferri-magnetic behavior which is affected by mechanical prestress. The ferrimagneticlattice consists of two interleaving lattices with contrary magnetization. For anapplied magnetic field H one of these lattices switches in the direction of H ,the other one remains unchanged. In the absence of mechanical loading bothlattices are switching. The specimen behaves like a ferromagnetic material.For mechanical prestress this switching process is hampered. As a result thecharacteristic hysteresis curves are separated in respectively two hysteresis asshown in Fig. 5.

2.2 Ferroelectric ceramics

The basic attribute of ferroelectric ceramics is the polarization, analog to themagnetization in ferromagnetics. Below Tc the centers of positive and negativecharges are situated at different locations in the unit cell, which results in aspontaneous polarization. The direction of the polarization can be changedby applying an electrical or mechanical loading. Due to the Coulomb forces,the central metal ion shifts within the unit cell which causes a distortion.Similar to ferromagnetism the direction of the polarization is aligned parallelon microscopic level. On the mesoscopic scale a domain structure evolvesto minimize the electrostatic energy. Again, the nonlinear behavior of thematerial is affected by switching processes along the domain walls resultingin wall movements. The observed hysteresis curves are very similar to the

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Figure 5: Ferromagnetic and butterfly hysteresis curves in magnetostrictivematerials

ones of magnetostrictive materials. Additionally, for a mechanical load cyclea so called ferroelastic hysteresis can be measured which does not occur inmagnetostrictives. Ferroelastic hysteresis effects are not considered in thispaper.

3 Governing equations

In this section the governing equations of magnetostrictive and piezoelectricmaterials are described. Again the first part is addicted to magnetostrictives.The annotations to piezoelectricity are summarized briefly with respect todifferences and similarities to the magnetoelastic coupling.

3.1 Magnetostriction

The balance equations of magnetostrictive coupled problems are the strongform of equilibrium and the Maxwell equation considering solenoidality of themagnetic field. The local forms are given as

S+ b = 0 (1)B = 0. (2)

Here, S stands for the stress tensor, b for the mechanical body forces andB for the magnetic flux density. Thus the influence of ponderomotive forcesis minor, they are neglected in this paper. In consequence the stress tensor issymmetric. The mechanical and magnetic boundary conditions read

S n = t on tB0 (3)B n = M on B0. (4)

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The vector t denotes the prescribed traction on the surface tB0 and the scalarvalue M the magnetic surface charge on B0. M is an imaginary quantity,because of magnetic monopoles do not exist. But its introduction is beneficialif the magnetic scalar potential is used, see also the examples in section 9. Thevector n gives the outward unit normal on B0. The linear strain tensor Eand the magnetic field strength H are defined as

E =1

2( u+ u) (5)

H = M , (6)where u is the displacement vector and M the magnetic scalar potential. Thescalar potential is used instead of the magnetic vector potential, because onlyfields without curl are considered. For the constitutive model the total strainsand the magnetic field strength are decomposed additively as

E = Er +Ei (7)

H = Hr + H i. (8)

The superscript r denotes the reversible part. The irreversible quantities, in-dexed by i, serve as internal variables and describe the dissipative part. Eq. (7)is in full accordance with small strain plasticity theory. The additive split offield strength is a novel aspect of the proposed model. More common is adecomposition of the magnetization, see Fang et al. (2004), or the split of thepolarization for piezoelectric models, see e. g. McMeeking and Landis (2002),Kamlah (2001), Schroder and Romanowski (2005). The additiv split of themagnetic field is an alternative approach to describe the remanent changes ofthe magnetization. It is not motivated physically but by the numerical treat-ment of the constitutive model. The analogy to plasticity permits the usageof soloing algorithms of computational inelasticity. Furthermore a Legendretransformation of the constitutive equations is not necessary for the finite ele-ment formulation. Fig. 6 illustrates H i in an idealized ferromagnetic hysteresiscurve. For a magnetic field lower Hc linear behavior is assumed. Accordingly, B < Mc with Mc = Hc is applied for the magnetic flux density. For fieldstrengths higher than Hc irreversible wall movements are presumed which areassociated with a change of the internal variable H i. For the fully magnetizedmaterial reversible behavior is assumed. The growth of H i stopped if its normis equal to H is. The saturation value H

is is estimated from Brem. Generally the

hysteresis is curved. Here the permeability , which is described by the gradi-ent of the tangent, depends in on the chosen working point. In the following is defined for H = 0 and B = Brem. Accordingly H is = Brem/ is alsovalid for the general case.

Due to conservation of volume, which is claimed for the irreversible strains,a one-to-one relation is assumed as

Ei =Es

( H is)2H i H i I. (9)

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Figure 6: Idealized ferromagnetic hysteresis

this is similar to ferroelectric models where this ansatz is common, see e.g.Kamlah (2001), McMeeking and Landis (2002). The one to one relation linksthe irreversible strains and field strength. This reduces the amount of internalvariables to the vector H i. But Eq. (9) is only valid for pure electrical loadings.Ferroelastic behavior or mechanical depolarization is not approximated withthis approach. The quantity I denotes the projection tensor of rank two givenas

I =3

2

(e e 1

31

)with e =

H i

H i . (10)

The magnetization arises along a given direction e. The formulation is re-stricted to an uniaxial constitutive model with this assumption. A reorienta-tion of the magnetization direction e is not possible yet. But the approachpermits a more efficient computational treatment of the model, see Sec. 5

3.2 Piezoelectricity

The balance equations of the piezoelectric coupled problem are given by Eq. (1)and the local form of the Gauss law

D = 0 (11)

instead of Eq. (2). The vector D stands for the dielectric displacements.In case of dielectrics free volume charges are not considered. The boundaryconditions are given in Eq. (3) and supplemented with

D n = E on B0, (12)

where E is the electric surface charge. The electric field E is described via

E = E (13)

11

with the electric scalar potential E. Similar to Klinkel (2006a) the totalelectric field is additively decomposed as

E = Er + Ei, (14)

where Ei is the irreversible electric field. It serves as internal variable andreplaces H i in Eq. (9). Additionally the saturation parameter Eis is applied

instead of H is and Pc instead of Mc. The similarities of magnetomechanical andelectromechanical coupling are obvious. Further the formulation is exclusivelyformulated for magnetostrictive problems.

4 Thermodynamic framework

The constitutive model is based on a free energy function which serves asthermodynamic potential

=1

2(E Ei) : : (E Ei) +

H i eH is

(E Ei) : H

12( H H i) ( H H i) + ( H i). (15)

Here, , , and are elasticity tensor, piezomagnetic coupling tensor andpermeability. In the present work a second order potential is used. ThusEq. (15) is quadratic in E und H . The last term ( H i) represents the

energy stored in the internal variable H i. With respect to plasticity it iscalled hardening function. According to the second law of thermodynamicsand with neglecting thermal effects the Clausium-Duhem inequality reads

D = S : E B H 0 (16)

The scalar D names the dissipated energy. Using standard arguments of ra-tional continuum mechanics, see Coleman and Noll (1963), the stress and themagnetic flux density can be derived as

S :=

E= : (E Ei) +

H i eH is

H (17)

B := H

=H i eH is

T : (E Ei) ( H H i). (18)

Applying this definitions and the derivation in time of Eq. (9) the dissipationreduces to

D = (2

Es

( H is)2

Ei: I H i +

H i

)

H i 0, (19)

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in which is defined as the work conjugated variable to the internal variableH i. The partial derivatives are given by

Ei= S (20)

H i=

1

H is

[(E Ei) : H

]e ( H H i)

H i. (21)

A switching criterion is introduced to signify the begin of irreversible domainprocesses

= M2c

1 0. (22)

Here the parameter Mc serves as coercive value. Reversible wall movements inthe domains are assumed for < 0. For irreversible wall movements it holds = 0. The postulate of maximum dissipation is formulated by a Lagrangefunctional with as constraint. It reads

L = D() + (), (23)

where is the Lagrange multiplier. The solution of Eq. (23) requires

L

= 0 andL

= () = 0. (24)

The necessary conditions for the existence of a local minimum are given bythe Kuhn-Tucker conditions

0, 0, = 0. (25)

The evolution equation for the internal variable H i is derived from Eq. (24)1

H i = 2

M2c. (26)

It gives the rate of H i for a given E and H . Eq.(19) implies that H i is

parallel to e a scalar valued internal variable = ( H i e)/ H is is introduced.Furthermore, its work conjugated variable is reduced to the scalar with = e which is given in Appendix B. The governing equations for theoptimization problem read

= 2

M2c(27)

=2

M2c 1 = 0. (28)

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4.1 Integration algorithm

The evolution equation (27) is integrated in time. Here the implicit Eulerscheme is used, because it is unconditionally stable. The time t is discretizedin time steps. It is assumed that all values for the time step tn are known. Theinternal variable n+1 at tn+1 = tn+t has to be determined for an incrementalE or H . The discrete form of Eq. (27) in the interval [tn, tn+1] read

n+1 = n + 2 n+1M2c

(29)

with = (tn+1 tn). For irreversible processes it holds n+1 = 0. Theswitching criterion depends on the conjugated variable n+1 which is a functionof the unknown n+1. For the first predictor step a trial value trial = n isassumed. If the switching criterion is satisfied with the trial value, trial isequal to n+1. If not, a radial return algorithm with the following residua isapplied

Ra = n+1 n 2 n+1M2c= 0

Rb =2n+1M2c

1 = 0.(30)

These equations are iteratively solved by a local Newton iteration

R(k)a

n+1+

R(k)a

= R(k)a

R(k)b

n+1+

R(k)b

= R(k)b

(31)

with(k+1)n+1 =

(k)n+1 +

(k+1) = (k) +.(32)

The iteration step is denoted by the superscript (k). The partial derivativesneeded in Eq.(31) are given in Appendix C.

4.2 Hardening function

The hardening function determines the shape of the ferromagnetic andbutterfly hysteresis curves. It is defined as

H i:=

[k H is + a arctanh

(b

)]e (33)

with the material parameters k, a, and b. The first summand means linearhardening. The second one restricts the growth of near saturation.

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5 Algorithmic consistent tangent moduli

The numerical solution of the boundary value problem requires an iterativemethod for nonlinear constitutive models. To obtain quadratic convergence theNewton-Raphson iteration scheme is applied. Here, consistent linearizationsof S and B are needed for the load step tn+1. It reads

dSn+1 = n+1 : dEn+1 + n+1 d Hn+1 (34)d Bn+1 = n+1 : dEn+1 + n+1 d Hn+1. (35)

The tangent moduli , n+1 and n+1 can be employed by numerical evalua-tion. A more efficient formulation is obtained by deriving them in closed form.The total derivatives of Sn+1 and Bn+1 are given by

dSn+1 =

SE : dE + S H d H + S dn+1

(36)

d Bn+1 =

BE : dE + B H d H + B

d

n+1

. (37)

Terms depending on the internal variable n+1 have to be eliminated. Theconsistency condition = 0 implies dn+1 = 0 for irreversible processes.Consequently dn+1 = 0 gives an equation for dn+1. It reads

dn+1 =

[

E: dE +

H d H

]1

n+1

. (38)

With substituting Eq. (38) in Eqs. (36) and (37) one obtains

dSn+1 =

[S

E 1

S

E

]: dE +

[S

H 1

S

H

] d H

n+1

(39)

d Bn+1 =

[ B

E 1

B

E

]: dE +

[ B

H 1

B

H

]d H

n+1

.

(40)

Evaluating the partial derivatives one observes the algorithmic consistent tan-gent moduli as

n+1 = 1

S

E(41)

n+1 = n+1 1

S

H(42)

n+1 = 1

B

H. (43)

The derivatives in Eqs. (36) to (43) are given in Appendix D.

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Figure 7: Idealized ferrimagnetic hysteresis with local description

6 Ferrimagnetic hysteresis effects

The presented constitutive model has to be extended to approximate ferrimag-netic hysteresis behavior. The modifications are:

Introduction of local magnetic field quantities Extended switching criterion

The ferrimagnetic hysteresis curve is divided in two separate curves in thefirst and third quadrant, see Fig. 7. For the approximation a local coordinatesystem is introduced. Its origin is in the center of the partial hysteresis. Thehorizontal offset is defined with the parameter H0. The local field quantitiesmarked with superscript L are defined as

HL = H H0 g e (44)HLi = H i + H is g e. (45)

The scalar valued variable g = sign( H e) gives the sign of the offset. The offsetB0 has not to be specified, because the magnetic flux density is determined bythe constitutive equations. The local quantities HL and HLi are applied fora local free energy function

L =1

2(E Ei) : : (E Ei) +

H i eH is

(E Ei) : HL

12( HL HLi) ( HL HLi) + ( HLi) (46)

16

which leads to local dissipation inequality

D = L HLi 0 (47)with

L = (2

Es

( H is)2

L

Ei: I ( HLi H is g e) +

L

HLi

). (48)

It is noted that H i = HLi H is g e is used in the one-to-one relation (9).The yield criterion is modified to provide reversible behavior in the global

origin H = 0. Here the sign of g changes. The modified yield criterion reads

L = sign( Mmax ML) L L( Mc)2

1 0 (49)

with Mmax = Mc + k His + a/arctanh(1/b). The vector

ML = ( HL HLi)denotes the uncoupled magnetization. For M > Mc the sign function isnegative and reversible behavior is ensured. With the Eqs. (46), (49), and thearguments of Section 4 the evolution equation

HLi = 2 sign( Mmax ML)L

Mc(50)

is developed. Analog to Section 4.1 the implicit Euler scheme is applied for theintergration of Eq. (50). The expanded model has no effect on the formulationof the algorithmic consistent tangent moduli of Section 5, because the signfunction vanishes with derivation.

7 Constitutive modelling of ferroelectric ce-

ramics

The presented constitutive model is applied almost unmodified for ferroelectricceramics. Here an additive split is done for the electric field with E = Er+ Ei.Furthermore, E and Ei replaces H and H i in the free energy function Eq. (15).

The switching criterion depends on the coercive polarization Pc. Due to thephenomenology of ferroelectric ceramics the expanded model of Section 6 isnot applied. However, near the Curie temperature ferroelectrics show a similarhysteresis behavior as ferrimagnetics, see Smith (2005).

8 Variational formulation and finite element

approximation

In this section the weak form of the coupled boundary value problem is derived.The boundary value problem is given by the balance equations (1) and (2),

17

the Neumann boundary conditions Eqs. (3) and (4), and Dirichlet boundaryconditions

u = u on uB0 (51)

= on B0. (52)

It holds B0 = tB0uB0, = tB0

uB0, B0 = B0

B0 and

= B0

B0. The weak form is obtained by multiplying Eqs. (1) and(2) with the test functions u and . Integration by part and application ofthe divergence theorem lead to

=

B0

E: E +

H H b u dV

tB0

t u dA

B0

M dA = 0.(53)

The virtual gradient fields are given by E = 12( u+ u) and H =

M . The space of the test functions is assumed asU = {u [H1(B0)]3 , u|uB0 = 0} (54)V = { H1(B0) , |B0 = 0}. (55)

8.1 Element formulation

The finite element approximation is based on the discretization of the domainB0 in nelem finite elements Be with B0 =

nelme=1 Be. The shape functions for

the presented hexahedral element read

NI =1

8

3i=1

(1 + iIi) with 1 i 1 (56)

and1I {1, 1, 1,1,1, 1, 1,1}2I {1,1,1, 1,1,1, 1, 1}3I {1,1,1,1, 1, 1, 1, 1} .

(57)

Due to the isoparametric concept the same shape function are used for theapproximation of the geometry X and the unknown field quantities u ,respectively. This leads to

Xhe =8

I=1

NIYI and dhe =

[uhehe

]= N ve. (58)

with N = [N1, N2, . . . , N8] and NI = diag[NI , NI , NI , NI ]. The vec-tor ve = [u

T1 , 1, u

T2 , 2, . . . , u

T8 , 8]

T consists of the unknown nodal

18

degrees of freedom. The nodal coordinates are given by the vector YI . Con-sidering the vector notation of Appendix A the approximated gradient fieldsread [

Ehe Hhe

]= B ve with B = [B1, B2, . . . , B8]. (59)

The matrices BI consist of derivations of the shape functions to the globalcoordinates Xi. It holds

BI =

[BIu 00 BI

]with BIu =

NI,X1 0 0

0 NI,X2 0

0 0 NI,X30 NI,X3 NI,X2

NI,X3 0 NI,X1NI,X2 NI,X1 0

and BI =

NI,X1NI,X2NI,X2

.(60)

The derivations NI,Xi are obtained with XNI = J1e NI . The matrix Jeis the Jacobian given by

Je = Xhe with = [

1,

2,

3]T . (61)

Considering the before mentioned interpolation the weak form on element levelis approximated by

he = vTe [

Be

BT[

E

H

]NT

[b0

]dV

Be

NT[tM

]dA]

Rhe

. (62)

After assembly over all elements one obtain the global weak form h =nelme=1

he with the global residual vector R

h =nelm

e=1 Rhe and the global nodal

degrees of freedom v =nelm

e=1 ve. Due to the nonlinear constitutive modelthe equation h = 0 is solved iteratively. The unknown node values vn+1 forthe load step tn+1 are estimated by a Newton-Raphson scheme. The residualvector Rh is expanded by a Taylor series which is truncated after the linearterm

Rh (K)n+1 +

Rh (K)

vn+1v

(K)n+1 = 0. (63)

After each iteration step (K) the nodal degrees of freedom are updated by

v(K+1)n+1 = v

(K)n+1 + v

(K)n+1. The start value is obtained from the last state of

equilibrium v(1)n+1 = vn. The approximations of the incremental gradient fields

[ETe , HTe ]

T = Bve are analog to Eq. (59). The tangent stiff matrix onelement level reads

[KeT ](K)n+1 :=

Rh (K)

vn+1=

Be

BTB dV

(K)

n+1

with (K)

n+1 =

[

T

](K)n+1

(64)

19

The algorithmic consistent tangent moduli , and for tn+1 are evaluatedwith Eqs. (41)-(43).

The element formulation for ferroelectric ceramics is based on an equiv-alent variational approach as applied for magnetostrictive alloys. Instead ofH the electric field strength E is used which is described with the electricscalar potential E, see Eq. (13). The local Gauss law (11) and the boundarycondition (12) replaces the solenoidality of the magnetic field and the cor-responding boundary condition, see Eqs. (2) and (4). This leads to a finiteelement approximation which differs only in the used field quantities from theone of Section 8.1.

9 Numerical simulations

The developed finite element formulation is implemented in the finite elementcode FEAP which is documented in Taylor (2008). In the following, differentexamples are used to show the capabilities of the proposed constitutive model.The first example deals with the nonlinear behavior of the magnetostrictive al-loy Terfenol-D. The occurring hysteresis curves are compared with experimen-tal data. An approximation of ferrimagnetic behavior is discussed in furthersimulations. In the second example a specimen consisting out of a ferroelectricceramic is analyzed. The results show that the model is able to reproduce thetypical hysteresis curves of ferroelectric ceramics correctly. In the third exam-ple an amplified magnetostrictive actuator is presented. To improve the designit is optimized in a parametric study. In the fourth example a piezoelectrichollow cylinder of soft lead lanthanum zirconate titanate is considered. Theborders of these cylinders tend to warp inwardly during polarization. Thiseffect will be discussed by means of the results. The fifth example deals witha piezoelectric telescope actuator. The results of the nonlinear simulation arecompared with experimental data.

9.1 Magnetostrictive hysteresis

A simple example is used to illustrate the ability of the presented constitutivemodel to account for magnetostrictive hysteresis phenomena. Here a cube ofTerfenol-D with an edge length of L = 20mm is considered. The boundaryand loading conditions are given in Fig. 8. The magnetic loading is set with thescalar potential M resulting in a homogeneous magnetic field in 3-direction.The field strength is oscillated in a zigzag function between H3 = 100 kA/mand H3 = 100 kA/m. The applied material parameters based on Dapino et al.(2006) and Moffett et al. (1991) are given in Tab. 1. The resulting hysteresiscurves compared with the experimental data of Engdahl (2000) are depictedin Fig. 9. Terfenol-D is a soft magnetic material. Thus, the range of linearbehavior is very small. Raising the magnetic field beyond the coercive valueMc results in an intense increase of B3 and E33 The growth slows down for

20

Figure 8: Boundary and loading conditions for the Terfenol-D specimen

Terfenol-DE1 = E3 = 29GN/m

2, 12 = 13 = 0.25, G12 = G13 = 11.6GN/m2

33 = 40.9Vs/Am2, 13 = 43.95Vs/Am2, 51 = 28.3Vs/Am2

1 = 2 = 3 = 2.8025 106 Vs/AmMc = 8.2 103 T, Eis = 1.08 103, H is = 185 kA/m k = 1.033a = 0.15/arctanh(1/b) T, b = 1.04

Table 1: Applied material parameters

values H3 > 50 kA/m, because of the arctanh-term in the hardening function.This leads to a saturation of the irreversible quantity for magnetic fieldshigher 80 kA/m. After unloading only low remanent values of B3 and E33are noticed. The ferromagnetic and butterfly hysteresis in Fig. 9 show goodagreement with the experimental data. It confirms that the model is able toapproximate the typical behavior of Terfenol-D.

A further aspect of this examples is the convergence behavior of the pre-sented constitutive model. For this, a load step from H3 = 5kA/m to H3 =11 kA/m is considered, which is within a range of high nonlinear growing. Theinterval is marked by dots in Fig. 9. The residua of the iteration processesare used to evaluate the convergence rates. The norms of the residuum vectorR of the global iteration are listed in the first column of Tab. 2. A typicallocal iteration is summarized on the right hand side. Therefore the residua ofEq. (30) are considered. It is noted that quadratic convergence is obtained forall residua.

Terfenol-D shows ferrimagnetic behavior for mechanical prestresses . Thepresented simulations account for different values of the parameter H0. The re-maining parameters are modified according to Tab. 3. The results are given inFig. 10. Depending on the offset H0 the separated hysteresis curves are shifteddifferently, which represents qualitatively the occurring material behavior. Asmentioned before the model does not include the influence of the mechanicalprestress. For this reason the curves are not compared with experimental data.

21

-1

-0.5

0

0.5

1

-100 -50 0 50 100

B3[T]

H3 [kA/m]

0

0.5

1

1.5

-100 -50 0 50 100

E33[10

3]

H3 [kA/m]

presented

Engdahl (2000)

Figure 9: Ferromagnetic and butterfly hysteresis compared to experimentaldata and the load step for convergence evaluation

iteration global iteration local iterationR Ra Rb

1 9.904572E + 01 7.196267E 02 1.052652E + 002 1.732041E + 02 3.088828E 02 9.670525E 023 7.862738E + 00 1.770798E 03 1.616275E 034 2.152431E 02 1.583016E 06 4.965618E 075 1.636638E 07 4.076455E 13 4.820135E 14Table 2: Convergence of the local and the global iteration process

9.2 Ferroelectric hysteresis

This examples is concerned with the simulation of ferroelectric ceramics. Acubic spezimen of lead lanthanum zirconate titanate (PLZT) is simulated. Theedge length is L = 10mm. The boundary conditions are analogue to Fig. 8 butE is used instead of M . The scalar potential E at the topside is oscillatingbetween 800V. The material parameters are given in Tab. 4. Fig. 11 showsthe resulting ferroelectric und butterfly hysteresis compared with the resultsof Hwang et al. (1995). A good agreement for both curves is observed. PLZThas a much higher coercive value than Terfenol-D which results in distinctive

Eis = 0.35 103, H is = 125 kA/m k = 1.13a = 0.1/arctanh(1/b) T, b = 1.05

Table 3: Modified material parameters of the hardening function for ferrimag-netic behavior

22

-1.5

-1

-0.5

0

0.5

1

1.5

-100 -50 0 50 100

B3[T]

H3 [kA/m]

0

0.5

1

1.5

2

-100 -50 0 50 100

E33[10

3]

H3 [kA/m]

H0 = 45 kA/mH0 = 55 kA/m

H0 = 65 kA/mH0 = 0

Figure 10: Ferrimagnetic and butterfly hysteresis for different parameters H0

PLZTE1 = E3 = 68GN/m

2 12 = 13 = 0.35, G13 = 25.19GN/m2

33 = 50.116C/m2, 13 = 14.96C/m2, 51 = 38.148C/m2

1 = 2 = 3 = 1.125C/kNm2

Pc = 0.405 103 C/m2 Eis = 1.44 103 Eis = 0.215 kV/m k = 0.9998 3a = 0.0005/arctanh(1/b) C/m2, b = 1.04

Table 4: Applied material parameters for PLZT

remanent values of D3 and E33.

9.3 Amplified magnetostrictive actuator

The displacements which can be achieved with magnetostrictive and piezo-electric materials are only small. Different approaches are used to increase theworking range of actuator devices. One possibility is a mechanical amplifiedactuator. An optimized design for such an device is discussed in the followingexample. The amplification is accomplished by a special frame, which increasesthe achieved displacements of the device, see Fig. 12.

The dotted outline represents the undeformed initial state. With applica-tion of a magnetic field the magnetostrictive element in the center changes itslength. The magnetic field is induced homogeneously by a surrounding coil,which is not shown in the plot. The contraction of the magnetostrictive rodcauses a movement w of the adapter plates. This movement is higher than thedeformation of the active material. In the following an own design is discussedwhich is based on an actuator of Energen, Inc. The used active material isTerfenol-D. The influence of the height of the frame on the mechanical proper-

23

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.8 -0.4 0 0.4 0.8

D3[C/m

2]

E3 [MV/m]

-0.5

0

0.5

1

1.5

2

2.5

-0.8 -0.4 0 0.4 0.8

E33[10

3]

E3 [MV/m]

presented Hwang et al. (1995)

Figure 11: Ferroelectric und butterfly hysteresis compared to experimentaldata

Terfenol-D

aluminumadapter

plate

Figure 12: Functioning of an amplified actuator

ties of actuator is examined in a parametric study. The aim is to determine theoptimal design for maximal amplitudes of displacement and force. The chosengeometry and finite element mesh are given in Fig. 13. The Terfenol-D rod hasa length of 50mm and a diameter of 14mm. With respect to symmetry onlyan eighth of the actuator is modeled with 654 hexahedral elements. This areais depicted white in Fig. 13. The simulations have shown that a further meshrefinement has no significant impact on the results. The displacements perpen-dicular to the symmetry plane are fixed at the symmetry borders. By settingthe scalar potential M = M at one end of the Terfenol-D rod a homogeneousmagnetic field in the rod is applied. The maximal field strength is 100 kA/m.The material parameters for Terfenol-D are given in Tab. 1. The character-istics of aluminum are modeled with E = 75000N/mm2 and = 0.33. The

deformed structure for H3 = 100 kA/m is depicted in Fig. 14. The influenceof the height h2 on the maximal displacements w is tested in the first part ofthe parametric study. The driven load cycle for H3 is 0, 100 kA/m, 0. Theheight h2 is varied between 38mm and 44mm. The resulting load displacementcurves are depicted in Fig. 15. The nonlinear material model causes hysteresis

24

Dimensions: l = 61mm, b = 30mm, h1 = 34mmt1 = 1mm, t2 = 2.2mm

Figure 13: Geometry and finite element mesh of the presented actuator

-0.030

-0.045

-0.060

-0.075

-0.090

-0.105

-0.120

Displacements u2 [mm]

Figure 14: Deformed structure for H3 = 100 kA/m und h2 = 41mm, thedisplacements are scaled by factor 30

effects for w, which affect in particular the range between 10 to 70 kA/m. Theamplitude of the hysteresis depends on h2. The maximal values of w can beachieved by devices with a height h2 = 41mm. Higher values of h2 decreasethe maximal displacements.

The second aspect of this example is the influence of the height h2 on theforces F2 which can be achieved with the actuator. In this case a working pointof H3 = 50 kA/m is considered. First of all the Terfenol-D rod is magnetizedwith a field strength of 100 kA/m. Then the load is reduced to the workingpoint. Now the adapter is fixed in w-direction at the current deformed positionand a further load cycle between 25 and 75 kA/m is driven. The reaction forcesF2 at the adapter plates are plotted in Fig. 16. Again hysteresis effects areobserved arising from the nonlinear constitutive model. An increasing heighth2 causes higher forces F2. But a heightening above h2 = 46mm has only low

25

-0.16

-0.14

-0.12

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0 20 40 60 80 100

w[m

m]

H3 [kA/m]

h2 = 38mm

h2 = 40mm

h2 = 41mm

h2 = 42mm

h2 = 44mm

Figure 15: Displacements w for different values h2

effect on F2.

26

-12

-10

-8

-6

-4

-2

0

2

4

6

8

20 30 40 50 60 70 80

F2[N

]

H3 [kA/m]

h2 = 38mm

h2 = 40mm

h2 = 42mm

h2 = 44mm

h2 = 46mm

h2 = 48mm

Figure 16: Forces at the adapter for different values h2

9.4 Piezoelectric hollow cylinder

A piezoelectric hollow cylinder of soft lead zirconate titanate (PZT) is consid-ered. The borders of these cylinders tend to warp inwardly during polariza-tion. This behavior will be discussed and the results will be compared withsimulations of Laskewitz et al. (2006). The geometry, boundary and loadingconditions are depicted in Fig. 17. With respect to rotational symmetry onlyone element is used in circumferential direction. The material parameters arechosen analog to Laskewitz et al. (2006), see Tab. 5. The local 3-direction ofthe material and the polarization direction e are parallel to radial directionr. Fig. 18 shows the deformed structure scaled by factor 30. In Fig. 18a) thecylinder is maximal loaded with E = 4.5 kV. Fig. 18b) depicts the structureafter unloading. In both cases a warping of the upper boundary inwardly isnoticed. This behavior can be explained with the logarithmic distribution ofthe electric potential in radial direction which results from the analytical solu-

27

:

Figure 17: Geometry and finite element mesh of the presented actuator

E1 = E3 = 0.06MN/mm2, 12 = 13 = 0.396, G13 = 0.02149MN/mm

2

33 = 21.796 106 C/mm2, 13 = 6.571 106 C/mm251 = 12.464 106 C/mm21 = 2 = 1.27796 109 C/MNmm2 3 = 0.74409 109 C/MNmm2Pc = 7.4409 109 C/mm2, Eis = 1.2 103, Eis = 38.9736MN/C, k = 1.00001 3a = 0.2 108 C/mm2, b = 1.0

Table 5: Applied material parameters for the hollow cylinder

tion of the governing Laplace equation 2E = 0. The electric field E3 at theinner boundary is higher than E3 at the outer boundary. This causes a higherpiezoelectric contraction inboards.

In Fig. 19 the normalized polarization || is plotted versus r. The results ofLaskewitz et al. (2006) are given for comparison. At full loading the materialis completely polarized across the entire width, see Fig. 19a). With unloadingthe outer area begins to depolarize as depicted in Fig. 19b). Here the presentedresults show a smoother transition compared to Laskewitz et al. (2006) whichis caused by the arctanh function in the hardening function. In Fig. 20 theferroelectric hysteresis are plotted at the discrete positions (I) and (II) markedin Fig. 19. The dotted part of the curve is the passed range. In (I) the electricfield rises much higher than in (II). Furthermore in (I) a positive electric fieldremains after unloading whereas a negative one is noticed in (II). This negativeelectric field causes the slight depolarization at the outer boundary. A similarbehavior is noticed for the relating butterfly hysteresis, see Fig. 21. Here thestrain component E33 and accordingly the transversal contraction are much

28

-0.001

-0.003

-0.005

-0.007

-0.009

-0.011

Displacements inz-direction [mm]

Figure 18: Deformed cylinder scaled by factor 30 compared with undeformedstructure: a) at maximal loading E = 4.5 kV, b) after unloading E = 0

higher in (I) than in (II). So we have a remaining warping of the cylinder afterunloading.

29

0.95

0.96

0.97

0.98

0.99

1.00

1.01

0.0 0.5 1.0 1.5

Polarization||

Radial Position r [mm]

a)

(I) (II)

0.95

0.96

0.97

0.98

0.99

1.00

1.01

0.0 0.5 1.0 1.5

Polarization||

Radial position r [mm]

b)

(I) (II)

presented

Laskewitz et al. (2006)

Figure 19: Normalized polarization r-direction: a) maximal loading E =4.5 kV, b) after unloading = 0

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-5 -4 -3 -2 -1 0 1 2 3 4 5

D3[C/m

m2]

E3 [kV/mm]

(I) r = 0.011mm

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-5 -4 -3 -2 -1 0 1 2 3 4 5

D3[C/m

m2]

E3 [kV/mm]

(II) r = 1.455mm

Figure 20: Ferroelectric hysteresis at (I) and (II)

30

0

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3 4 5

E33[10

3]

E3 [kV/mm]

(I) r = 0.011mm

0

1

2

3

4

-5 -4 -3 -2 -1 0 1 2 3 4 5

E33[10

3]

E3 [kV/mm]

(II) r = 1.455mm

Figure 21: Butterfly hysteresis at (I) and (II)

31

9.5 Piezoelectric telescope actuator

Telescope actuators are a common design for actuators. They consist of con-centric cylinders connected by end-caps. The idea of the design is an alternat-ing contraction respectively extension of the cylinders by an electric loadingto amplify the achievable displacements. In this example a telescope actua-tor fabricated by injection molding is discussed. The design is presented byAlexander et al. (2001). The geometry with dimensions and outer diametersare depicted in Fig. 22. With respect to symmetry only a quarter of 90

is meshed with 430 hexahedral elements. The examined displacement is w.

Dimensions in mm, wall thickness = 1mm

Figure 22: Geometry and finite element mesh of the telescope actuator

The applied material parameters for the ferroelectric ceramic MSI-53HD aregiven in Tab. 6 and are estimated on the basis of Alexander et al. (2001).The privileged 3-direction of the material points is the radial direction. Un-like the conventional manufactured actuators the injection molded design ispolarized after fabrication. To polarize the cylinders a radial electric field ofE3 = 1200V/mm is applied. Later on an electric field of E3 = 300V/mm isused for the working mode. The electric potential is applied by nickel elec-trodes as depicted in Fig. 23a). The resulting deformation scaled by factor100 is given in Fig. 23b). The displacement w is plotted across the electric

field for the loading cycle E3 = 0, 300V/mm, -300V/mm, 0. The results aredepicted in Fig. 24. Due to the nonlinear constitutive law hysteresis effects areobserved in the plot, which show good agreement with the experimental resultsof Alexander et al. (2001). The linear analytical solution of Alexander et al.(2001) can not reproduce hysteresis effects. It is noted that the presentedresults are moved by 9.545m to account for the deformation following fromthe polarization process. The nonlinear behavior results from a depolariza-

32

E1 = 60606N/mm2, E3 = 48310N/mm

2, 12 = 0.290, 13 = 0.408G13 = 22990N/mm

2

33 = 10.63 106 C/mm2, 13 = 29.88 106 C/mm2, 51 = 27.26 106 C/mm21 = 2 = 3 = 1.0693 1014 C/Nmm2Pc = 0.7693 108 C/mm2

Eis = 1.2 103, Eis = 17.63507V/mm, k = 1.006 3a = 0.0099 106/arctanh(1/b) C/mm2, b = 1.0005

Table 6: Applied material parameters for MSI-53HD

tion during the loading cycle. In Fig. 25a) the normalized polarization in the

cylinders is plotted for E3 = 300V/mm. In the cylinder 2 and 4 the elec-trical field points opposite to the direction of polarization. As a result thepolarization decrease to 95%. For a loading in the opposite direction withE3 = 300V/mm the remaining cylinders are depolarized, see Fig. 25b). Thisdepolarization of the cylinders is completed after one load cycle. A recurringminor hysteresis obtained in Alexander et al. (2001) can not be approximated,since this behavior is not considered within the constitutive model.

-0.15 -0.05 0.05 0.15 0 2 4 6 8 10 12 14 16

a) b)

Voltage U [V] Displacements w [m]

Figure 23: Energized actuator, E = 300V/mm: a) loading condition b) de-formed actuator, displacements scaled by factor 100

10 Conclusion

In this paper a nonlinear constitutive model for magnetostrictive and piezo-electric materials is presented. Beside the general approach an additive split

33

-15

-10

-5

0

5

10

15

-400 -300 -200 -100 0 100 200 300 400

displacementw

[m]

electric field E [V/mm]

experimental results from Alexander et al. (2001)

analytical solution from Alexander et al. (2001)

presented

Figure 24: Hysteresis behavior of the telescope actuator in comparison withAlexander et al. (2001)

of the field strength in a reversible and an irreversible part is a novel aspectof the proposed macroscopic model. The irreversible part serves as internalvariable describing the domain state of the material. The additive split ofthe field strength allows a consistent finite element approximation employingdisplacements and scalar potential as independent variables. The irreversiblestrains which arises from domain switching processes are determined by theirreversible field strength with a one-to-one relation. The model is embeddedin a thermodynamic consistent framework, which is based on the definition ofa free energy function and a switching criterion. It is able to reproduce theferroelectric and butterfly hysteresis of ferroelectric ceramics. For the simu-lation of magnetostrictive materials an expansion is proposed which includesferrimagnetic behavior. The model is incorporated in a finite element formu-lation. The algorithmic consistent tangent moduli are given in closed form. Inthe presented numerical examples the results are compared with experimentaldata, which demonstrate the capability of the formulation.

34

0.05

-0.15

-0.35

-0.55

-0.75

-0.95

Polarization

a) b)

Figure 25: Polarisation in the actuator: a) E = 300V/mm b) E = 300V/mm

A Vector notation and linear material con-

stants

Vector notation of the strain and stress tensor:

E = [E11, E22, E33, 2E23, 2E13, 2E12]T (65)

S = [S11, S22, S33, S23, S13, S12]T (66)

Elasticity tensor in matrix form:

1 =

1E1

12E1

13E3

0 0 0

12E1

1E1

13E3

0 0 0

13E3

13E3

1E3

0 0 0

0 0 0 1G13

0 0

0 0 0 0 1G13

0

0 0 0 0 0 1G12

(67)

with G12 =E1

2(1+12).

Piezomagnetic respectively piezoelectric couple tensor in matrix form, mag-netic permeability and electric permittivity:

=

0 0 130 0 130 0 330 51 051 0 00 0 0

, =

1 0 00 2 00 0 3

, =

1 0 00 2 00 0 3

(68)

35

B Scalar valued conjugated variable to the in-

ternal variable

= I :[ : I 23 Es H 32 : E 2

] EsH is

H T : [E I Es2] 1H is e ( H e H is )

H i e H is (69)

= 2 Es

H isI :

[ : I 32Es H 3 : E

]+ e e H is

H i e H is (70)

(71)

C Derivatives for the integration algorithm

R(k)a

n+1= 1 2

M2c

n+1n+1

(72)

R(k)a

= 2 n+1

M2c(73)

R(k)b

n+1=

2

M2c

n+1n+1

(74)

R(k)b

= 0 (75)

36

D Derivatives for the algorithmic consistent

tangent moduli

S

= : I 2Es+ H (76)

B

= T : E T : I 3Es2 + e H is (77)

E= : I 2

EsH is

H 1H is

(78)

H= T : [E 3Ei] 1

H is e (79)

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Alexander, P., Brei, D., Miao, W., Halloran, J., Gentilman, R., Schmidt, G.,McGuire, P., Hollenbeck, J., 2001. Fabrication and experimental characteri-zation of d31 telescopic piezoelectric actuators. Journal of Materials Science36 (17), 42314237. 9.5, 9.5, 9.5, 9.5, 24

Bozorth, R., 1964. Ferromagnetism, 8th Edition. Van Nostrand. 2

Butz, A., Klinkel, S., Wagner, W., 2008. A piezoelectric 3d-beam fi-nite element formulation accounting for geometrical and material non-linearities. International Journal for Numerical Methods in EngineeringDOI:10.1002/nme.2320. 1

Butz, A., Klinkel, S., Wagner, W., 2005. A nonlinear piezoelectric 3d-beamfinite element formulation. In: Bathe, K.-J. (Ed.), Proceedings Third MITConference on Computational Fluid and Solid Mechanics. MassachusettsInstitute of Technology, Cambridge, pp. 291296. 1

Calkins, F., Smith, R., Flatau, A., 2000. Energy-based hysteresis model formagnetostrictive transducers. Magnetics, IEEE Transactions on 36 (2), 429439. 1

Carman, G., Mitrovic, M., 1995. Nonlinear constitutive relations for magne-tostrictive materials with applications to 1-d problems. Journal of IntelligentMaterial Systems and Structures 6 (5), 673683. 1

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ContentsIntroductionPhenomenological behaviorMagnetostrictive materialsFerroelectric ceramics

Governing equationsMagnetostrictionPiezoelectricity

Thermodynamic frameworkIntegration algorithmHardening function

Algorithmic consistent tangent moduliFerrimagnetic hysteresis effectsConstitutive modelling of ferroelectric ceramicsVariational formulation and finite element approximationElement formulation

Numerical simulationsMagnetostrictive hysteresisFerroelectric hysteresisAmplified magnetostrictive actuatorPiezoelectric hollow cylinderPiezoelectric telescope actuator

ConclusionVector notation and linear material constantsScalar valued conjugated variable to the internal variableDerivatives for the integration algorithmDerivatives for the algorithmic consistent tangent moduli