The Multiplicative Decomposition in Continuum Mechanics applied to Smart Structures

In the present paper, the concept of multiplicative decomposition of the deformation gradient into an elastic and an inelastic part is adopted for the modelling of the nonlinear material behaviour of piezoelectric continua. Within this approach, the constitutive equations can be split into a purely elastic relation, as well as a relation accounting for the electrically induced part of the deformation. As a contribution to smart structure theories, the decomposition is used to include piezoelectric behaviour in Reissner's geometrically exact relations for the plane deformation of a Bernoulli-Euler beam, for which a continuum mechanics interpretation of the strain measures and stress resultants has been found recently. This allows us to incorporate the continuum mechanics decomposition approach consistently into the beam theory. Finally, a variational formulation is presented, from which the local balance equations can be obtained, and which may be used for the finite element discretisation of a smart beam.