Keywords: piezoelectricity, multiplicative decomposition, intermediate configuration, constitutive modelling, smart structures, beam theory.

The key idea in this paper is the notion of a stress-free intermediate configuration, which emerges from the multiplicative decomposition of the deformation gradient into an elastic as well as an inelastic part. This fundamental concept in the description of inelastic material behaviour in nonlinear continuum mechanics [1] is applied to the modelling of piezoelectric beams.

To this end, the stress resultants and kinematic relations of Reissner's geometrically exact theory for the plane deformation of beams [2] are presented and the interpretation of these quantities in terms of stress and strain tensors of nonlinear continuum mechanics is discussed [3]. Due to this continuum mechanics foundation, Reissner's beam theory is particularly interesting since it allows us to incorporate the concept of the multiplicative decomposition consistently.

The balance equations of piezoelectric continua are presented in the material representation along with the boundary conditions. The balance of the Helmholtz free energy is needed later in order to obtain the constitutive equations.

The multiplicative decomposition of the deformation gradient into an elastic and a piezoelectric part is discussed. The general concept is specified for the kinematic assumptions of the Bernoulli-Euler hypothesis, for which the elastic, piezoelectric and total strain tensors only have a single non-vanishing component.

In this approach, the free energy is split additively into a purely elastic part, for which any strain energy function from the elasticity theory can be chosen, and a piezoelectric part, which is identified from comparison with a free energy function of nonlinear piezoelectricity in the literature [4]. The constitutive relations for the stress tensor and the polarisation vector are derived using the assumption that the piezoelectric stretch depends linearly on the electric field. It is shown that these nonlinear equations may be linearised for most technical applications. The material parameter which is introduced in the linear approximation of the piezoelectric stretch is identified by comparison with the linear theory of piezoelectricity.

Finally, a variational formulation is derived by introducing a functional over the displacement field and the electric field. It is shown that the balance equations follow from it and the reduction for piezoelectric actuation is presented. Using the interpretation of Reissner's virtual work of the internal forces with the continuum mechanics equivalent, a suitable form for a finite element discretisation is obtained.

1

V.A. Lubarda, "Constitutive theories based on the multiplicative decomposition of defeformation gradient: Thermoelasticity, elastoplasticity, and biomechanics", Appl. Mech. Rev., 57(4), 2004. doi:10.1115/1.1591000

2

E. Reissner, "On one-dimensional finite-strain beam theory: The plane problem", Z. Angew. Math. Phys., 23(5), 795-804, 1972. doi:10.1007/BF01602645

3

H. Irschik, J. Gerstmayr, "A continuum mechanics based derivation of Reissner's large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli-Euler beams", Acta Mechanica, 206(1-2), 1-21, 2009. doi:10.1007/s00707-008-0085-8

4

J. Yang, "An Introduction to the Theory of Piezoelectricity", Springer, 2005.

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