Romanowski, H.; Schroder, J.

On the modelling of nonlinear electro-mechanical coupled ferroelectrica

In this contribution we discuss a simple phenomenological model for an assumed transversely isotropic ferroelectriccrystal. We focus on the theoretical formulation and numerical treatment of the coupled electro-mechanical behaviour.The main goal of this investigation is to capture the main characteristics of nonlinear ferroelectrica, such as thepolarization-electric-field and the quadratic strain-electric-field (butterfly) hysteresis loops. The history dependencyis modelled via two internal variables, the irreversible polarization and its conjugated irreversible electric-field.

1. Introduction

A characteristic feature of ferroelectric crystals is the appearance of a spontaneous polarization whose directioncan be affected by an applied electric field, see e.g. [8] and [2]. This ferroelectric phase transition occurs at theso-called coercive field Ec. This quantity depends on the loading history and has a maximum value Ec,max at highelectric-fields. A discussion of nonlinear ferroelectric dissipative effects for one-dimensional problems can also befound in [4]. A phenomenological model of ferroelectrics for general loading histories has been proposed in [3]. Foran introduction to piezoelectrica, see e.g. [5] and the references therein.

2. Constitutive Model

The governing field equations for the quasi-static case are the equation of momentum div[σ] + f = 0 and the Gaussequation div[D] = 0. Here σ represents the symmetric Cauchy stress tensor, f is the given body force and D is thevector of electric displacements. The electric displacements are split into D = ε0E +P , where ε0 is the permeabilityin vacuum, E the electric-field and P the polarization. The latter quantity is split into a reversible P r and anirreversible part P i. Electrical properties are strongly correlated to the structure of the crystal and the axis of theso-called spontaneous polarization is usually a crystal axis. Spontaneous polarization is defined by the value of thecharge per unit area on the surface perpendicular to the polarization axis. For a brief discussion of this topic seee.g. [7]. In the following we restrict our considerations to an ideal transversely isotropic material with the preferreddirection a with ||a|| = 1. Based on the principle of superposition of symmetries we arrive at the constitutiverelationships for the stresses and electric displacements which can be formulated as isotropic tensorfunctions. For anintroduction to, and the application of, the invariant theory see [1] and for the application of piezo-electric materialssee [6]. The invariants of interest are I1 := trace[ε], I4 := trace[ε(a ⊗ a)] and J2 := trace[(E ⊗ a)]. For an explicitexpression of the constitutive equation of D we assume that

D = −2γ1E − 2γ2J2a| J i

2 |Ec

− [β1I1a + β2I4a + β3aε]J i

2

Ec+ P i . (1)

Here J i2 represents the projection of the internal variable Ei , the conjugate to P i, onto the preferred direction, i.e.

J i2 := Ei · a. Also the stress function is assumed to depend on the internal variable J i

2 in the form

σ = C : ε + [β1J21 + β2J2(a ⊗ a) +12β3(E ⊗ a + a⊗E)]

J i2

Ec, (2)

with the fourth order elasticity tensor C. The invariant formulation of C with respect to the full basis is given in [6]and omitted here. The evolution of the internal variable Ei occurs if the loading function

φ(E, α) = | J2 | −α ≤ 0 with J2 = sgn(J2) min{| J2 |, Ec,max} and α :=| J i2 |≤ Ec,max (3)

is violated. In the loading case φ = 0, we assume that the evolution equation Ei

= λ ∂Eφ = λ sgn(J2) a. Theevolution of the irreversible polarization is governed by

Pi ∼ −λ ∂Eiφ = J i

2

sgn(J i2)

sgn(J2)a with the assumption P

i= f(J i

2) J i2 PS a , (4)

PAMM · Proc. Appl. Math. Mech. 3, 216–217 (2003) / DOI 10.1002/pamm.200310382

where f(J i2) is part of the integrand of a function f :=

∫ t

0 f(J i2) J i

2 dt ∈ [−1, 1] with f(J i2) = −f(−J i

2) where themaximum value of P i in a-direction, the saturation value of the polarization P S , is reached at α = Ec,max.

3. Algorithmic Treatement

The evaluation of the constitutive functions of the internal variables is carried out within a backward Euler-scheme.After computing the internal polarization and the internal variable J i

2, which controls the polarization inducedanisotropy, we can evaluate the constitutive equations (1) and (2). For the solution of the associated weak forms ofthe governing field equations with the Finite-Element-Method we apply a standard Newton iteration scheme whichrequires the consistent linearization of the weak forms in order to guarantee the quadratic convergence rate nearthe solution. Here, we focus only on the linearization of the coupled nonlinear constitutive equations. Formally wecompute the linear increments

∆σ = ∂εσ : ∆ε + ∂Eσ ·∆E + ∂Eiσ ·∆Ei =: C : ∆ε− eTσ ·∆E

∆D = ∂εD : ∆ε + ∂ED ·∆E + ∂EiD ·∆Ei =: eD : ∆ε + ε ·∆E ,(5)

with the generalized tangent moduli C, −eTσ , eD and ε. For the saturated state, i.e. α = Ec,max and E

i= 0 we

obtain the well-known relation of linear piezoelectricity e := eD = eTσ , with the definition of the transpose operator

of the third order tensor eTijk = ekij . The material parameters appearing in the invariant setting can directly be

identified with the classical coordinate-variant representation, see [6].

4. Numerical Example

As a simple example we consider a homogeneous strip with the referred direction a = (1, 0, 0)T and the dimensionslx = 2 mm and ly = 1 mm. At the left vertical boundary we prescribe an electric potential of 0.0 Volt and at theright vertical boundary a cyclic varying potential φ(t). For simplicity we set f(J i

2) = 1. The electric-field E1(t)versus time is depicted in Figure 1a. The obtained minor and maximal polarization-electric-field hysteresis loop andthe strain-electric-field butterfly hysteresis loop are depicted in Figure 1b and 1c, respectively.

-400

-300

-200

-100

0

100

200

300

400

0 2 4 6 8 10 12 14 16 18-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5

-0.5

0

0.5

1

-1.5 -1 -0.5 0 0.5 1 1.5a) b) c)

E1/Ec,max

P1/PS

E1/Ec,max

ε11/ε11,max

Figure 1: a) Loading E1(t) versus time b) P -E hysteresis loop c) ε-E hyteresis loop

5. Conclusion

In this contribution we have proposed a simple model for the description of ferroelectric phase transitions in an idealtransversely isotropic solid and fixed polarization axis. The main characteristics, the P −E hysteresis loop and theε− E butterfly loop of an ideal crystal, are reproduced very well.

6. References

1 Boehler, J.P. Representations for Isotropic and Anisotropic Non-Polynominal Tensor Functions. CISM Course No. 292.Wien-New York: Springer (1987).

2 Eringen, A.C.; Maugin, G.A. Electrodynamics of Continua, Vol. 1 & Vol. 2. New York: Springer (1990).3 Kamlah, M.; Tsakmakis, C. Phenomenological modeling of the non-linear electro-mechanical coupling in ferroelectrics.

Int. J. Solids Struct. 36 (1999), 669-695.4 Maugin, G.A.; Pouget, J.; Drouot, R.; Collet, B. Nonlinear electromechanical couplings. New York: Wiley (1992).5 Nowacki, W. Foundations of Linear Piezoelectricity. CISM Course No. 257. Wien-New York: Springer (1979).6 Schroder, J.; Gross, D.: Invariant Formulation of the Electro-Mechanical Entalpy Funktion of Transversely Isotropic

Piezo-Electric Materials. submitted to Arch. Appl. Mech. (2002).7 Xu, Y.: Ferroelectric Materials and their Applications. North-Holland: Elsevier Science (1991).8 Zheludev, I.S.: Physics of Polycrystalline Dielectrics Vol.2, Electrical Properties. New York: Plenum Press (1971).

Dipl.-Ing. H. Romanowski, Prof. Dr.-Ing. J. Schroder,Institut fur Mechanik, FB10 Bauwesen, Universitat Duisburg-Essen, Universitatsstr. 15, 45141 Essen.

m#`dgn] [@ ~V#nBZ n4 L>d#Bg>;S 217