Micromorphic Electroelastic Theory And Its Application To Piezoelectric Materials In Micro Scale

Piezoelectric ceramics has been widely used in sensors, actuators and transducers due to its intrinsic property of electromechanical coupling. With the development of nano technology and industrial evolution, more and more microminiaturized devices are produced, which makes an urgent demand for the piezoelectric analyses in nano-microscale. However, many novel phenomena, including size effect, non-local effect and high nonlinearity, were found. They exhibit the inconsistencies between the experimental observation and theoretical predictions from classical continuum theories. Therefore, it is necessary to develop a new theoretical method in order to realize life prediction and optimal design of piezoelectric devices in nano-microscale.As a natural extension of the micromorphic continuum theory, the linear theory of micromorphic (thermo-) electroelasticity is developed to characterize the nano-microscale behaviors of piezoelectric materials with remarkable microstructures. Families of generalized variational principles for linear static micromorphic electroelasticity and electroelastodynamics systems are deduced. For applications, the theory is used to describe the size effects in bending piezoelectric micro-beams and the anti-plane problems of piezoelectric media with a circular void or inclusion. The main results are shown as follows:(1) The linear theory of micromorphic electroelasticity is established by incorporating the coupled electromechanical behavior into the framework of micromorphic continuum theory. The families of generalized variational principles for linear micromorphic electroelasticity including the minimum potential energy principle, the minimum complementary energy principle, the generalized Hu-Washizu principle, the generalized Hellinger-Reissner principle are deduced. Furthermore, the Gurtin-typed generalized variational principle for mixed boundary-initial value problems of micromorphic electroelastodynamics are also obtained.(2) Based on the micromorphic electroelastic theory, the constitutive relations for micromorphic thermoelectroelasticity are deduced by means of the theorem of energy equivalence from the standpoint of thermodynamics. All of the reciprocal theorem, the generalized variational principle and the generalized Hamilton principle for the mixed boundary-initial value problem in convolution form are established. As an example, steady state responses of an unbounded homogeneous and isotropic micromorphic thermoelectroelastic body to external concentrated loads of mechanical, electric, and thermal origins are solved, respectively.(3) Referring to the work by Forest et al, a modified theory of micromorphic electroelasticity is presented in order to describe size effect in bending piezoelectric micro-beams. The governing equations as well as the boundary conditions are derived through the variational principle on some hypothetical preconditions. The solutions of the cantilever piezoelectric beam loaded by a concentrated force at the free end and the simply supported beam under a distributed load are obtained. As a result, it is observed that the size effect is significant only when the beam thickness is comparable to the characteristic length scale parameter.(4) The modified theory of micromorphic electroelasticity is further extended to solve the anti-plane problems of piezoelectric media. The electro-mechanical field solutions for a transversely isotropic piezoelectric medium are derived in the context of micromorphic electroelasticity. A generalized characteristic length is introduced in the formulation to describe the size effect. For illustration, an infinite piezoelectric medium containing a circular void or piezoelectric inclusion is considered. Numerical results reveal that the mechanical and electric fields predicted by the present model highly depend on the relative size of the void or inclusion with respect to the generalized characteristic length of the material, which is different from the classical prediction. The effects of the material parameters on the distribution of the mechanical and electric fields at micro scale are also discussed.The above work indicate that the theoretical model for micromorphic electroelastic continuum established in this paper is capable to describe the size effects of the mechanical-electrical coupling behaviors for piezoelectric materials at micro scale, which has important guiding significance on the prediction and optimal design of piezoelectric devices in microscale. This work also lays a base for the further development of micromorphic continuum and other generalized continuum theories.