A Couple Stress Theory Including Flexoelectric Effects For Transversely Isotropic Piezoelectric Materials

As the main component of MEMS,piezoelectric components have significant flexoelectric effects and size effects in addition to the piezoelectric effect at the micro/nano scale.In order to explain this phenomenon effectively,advanded theories such as strain gradient theory and the couple stress theory have been developed.However,the current theories mostly study isotropic materials in order to grasp the commonality of micro-nano scale structures when they are used to solve practical problems.When some scholars are studying the piezoelectric materials,in order to simplify the calculation,only the piezoelectric coefficient is considered to be anisotropy,while the other material coefficients are considered to be isotropic.Some scholars combine the piezoelectric effect with the flexoelectric effect as an equivalent piezoelectric effect.Some scholars separate the piezoelectric effect alone to study the effect of flexoelectricity.However,some scholars have found that piezoelectric materials under polarization conditions are transversely isotropic.As a special anisotropic material,a transversely isotropic material has higher symmetry.If the anisotropic theory is used to solve it,the calculation process is too complicated.Compared with isotropic materials,there are many worthwhile features to study.Therefore,it is completely necessary to separate the transversely isotropic material from the general anisotropic material and study it exclusively.In this paper,based on the Mindlin couple stress theory in which the rotation gradient is dependent on displacement,the constitutive equations,equilibrium equations and boundary conditions of the couple stress theory considering the effects of rotational gradient and polarization gradient are deduced.All the basic tensors included in the constitutive equations are subjected to transverse isotropic decomposition.And then the transverse tensors of of their coupling coefficient tensors are obtained by the construction of two transverse tensors.and the theory of transversely isotropic couple stress of the piezoelectric material is obtained.Using the newly developed transversely isotropic couple stress theory,basing on the plane strain hypothesis,the bending theory of piezoelectric cantilever beam was developed.And the analytical solution of it including the flexoelectric effect,piezoelectric effect,polarization gradient and rotational gradient has been solved.The analytical solution,using the degree of agreement between the theoretical degeneration result and the traditional theory,indirectly verifies the validity of the theory.The in-depth analysis of the generalized electrostatic stress term is discussed,and the reasonable basis for simplification is discussed.Finally,an example is given to calculate the force-electric coupling characteristics of a transversely isotropic piezoelectric beam subjected to electrical loads.The results show that the effect of generalized electrostatic stress can be omitted in the Euler Bernoulli cantilever piezoelectric beam subjected to electrical loads,both on the micrometer scale and on the nanometer scale.The positive and negative flexoelectrical effects of cantilever beams all show obvious size dependence,and weaken with the decrease of the flexoelectrical coupling coefficient.Especially,when the flexoelectric coupling coefficient is zero,the positive and negative flexoelectric effects disappear.The flexoelectric effect has a significant effect on the piezoelectric effect.This conclusion can guide the use of experiments to estimate the material feature size,which is of great significance.The transversely isotropic couple stress theory presented in this paper can effectively predict the size effect of the mechanical properties of micro-components and the flexoelectric effect of the force-electric coupling performance.It can effectively explain the mutual influence mechanism between the flexoelectric effect and the piezoelectric effect.It has an important guiding significance to the basic research of stress theory,MEMS product design,performance prediction and experimental research.