In a consistent way, the kinematical relations of Reissner's beam formulation are utilized in order to derive consistent definitions for bending moments, normal forces and shear forces, which are presented in terms of components of the second Piola Kirchhoff stress and the Green strain.

As an important result, any constitutive model formulated in terms of relations between stresses and strains can be employed in Reissners theory, taking into account the geometrical restrictions given in the shear deformable beam element. The present theory circumvents the common problem that Reissners shear deformable beam theory was originally formulated at the cross-sectional level only, deserving constitutive relations between bending moments, normal forces and shear forces and certain generalized strains.

In extension of previous works of the authors, the continuum mechanics based formulation is extended with respect to the effect of piezoelectric actuation. The piezoelectric actuation is taken into account by assuminge that the electric field vector in the actual configuration is directed perpendicular to the deformed axis, and the pull-back of this quantity is considered in the Reissner-type formulation. The piezoelectrically actuating stress resultants are derived and can be used both, within classical Reissners theory as well as within the continuum mechanics formulation. As a figurative example, a cantilever beam under a tip couple is studied in order to discusse the effect of this derivation of the piezoelectric actuation in comparison to commonly used linear simplified relations.

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E. Reissner, "On one-dimensional finite-strain beam theory: the plane problem", Journal of Applied Mathematics and Physics (ZAMP), 23, 795-804, 1972. doi:10.1007/BF01602645

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