In this chapter, we start with a review on the classical treatment of electromechanics in a continua and an introduction of Toupin’s variational principle, which recovers the classical theories. Using this principle and extend it to include strain gradients, we are able to derive the governing equations and boundary conditions for general flexoelectric dielectrics. We then propose a linear constitutive law and prove a reciprocal theorem. An analogous theorem for piezoelectric materials is well known. Following that, we specialize to the study of isotropic materials and derive the governing Navier equations for the problem. Intrinsic length scales are introduced into the governing differential equations to raise the order of these equations. These intrinsic length scales are also different from the SGE ones due to the coupling effect. To ensure the uniqueness of the governing differential equations, an
We have also used our theory to solve some boundary values problems for isotropic flexoelectric materials relevant to experiments and application. First we look at a flexo- electric beam. We give expressions for effective bending flexoelectric constant, which is a linear combination of transverse and longitudinal flexoelectric constant. A beam equation is derived where electric voltage will generate extra bending moment proportionally. Bending stiffness/rigidity is different in short and open circuited beams due to the coupling effect and that difference is size-dependent. Then we turn to the torsion of a cylindrical speci- men but only to find that in isotropic and cubic materials, this deformation field will not create any flexoelectric coupling. Last we look into the hollow cylinders. Radially loaded cylinders offer insights into how the mechanical behavior of a flexoelectric material can be modulated at the nanoscale by the use of electric fields and vise versa. It demonstrates a way to control stress concentration by use of flexoelectric materials. Circumferentially sheared cylinder creates a unique azimuthal polarization field. Of course, there are many other problems that are of interests.
The methods discussed in this chapter is the starting point of studying flexoelectric solids, which we will use for the rest of the dissertation. It is also the fundamental of building continuum based computational methods for flexoelectric solids. Such methods will be required to compute displacement and polarization fields in complex geometries where close form solutions are not possible. Our solutions to various problems are useful in the interpretation of nanoscale experiments in the burgeoning field of flexoelectric materials. They are also useful in terms of providing benchmark problems to validate computational methods for flexoelectric solids.
Chapter 3
Defects in Flexoelectric Solid
3.1
Introduction
In this chapter, we will utilize the continuum framework in Chapter 2 to describe the stress and polarization fields near defects in flexo-electric solids. Defects are the spots where the effects of flexo-electricity are expected to be prominent due to the large strain gradients in their vicinity. As far as we know, there has been very few theoretical work that deals with this topic directly, but there is definitely a surging interests into it. In a computational study by Ong & Reed (2012), it is shown that overall piezoelectric behavior can be achieved by adopting defects into centrosymmetric graphene. This idea was later realized by Zelisko et al. (2014). According to their report, it is the interplay of non-centrosymmetric defects and flexoelectricity that creates this phenomenon. This indeed opens up the possibility of manipulating materials by flexoelectricity.
Flexoelectricity can also interplay with dislocations and generate many interesting phe- nomena. One example could be the experiments by Koehler et al. (1962), Turch´anyi, G. et al. (1973), Whitworth (1975), as mentioned in the Introduction. They studied charged dislocations in cubic crystals, e.g. alkali halides. These solids have centrosymmetric lat- tices which rule out piezoelectricity as the cause for the charge carried by dislocations in them, but this symmetry does not rule out flexoelectricity. In fact, flexoelectric phenomena can be observed in dielectrics of any symmetry group, including isotropic ones. Charged dislocations were also observed in experiments on ice by Petrenko and co-workers in 1980s,
like in Petrenko & Whitworth (1983). They conducted a thorough study of the electrome- chanical properties of ice and attributed charged dislocations and other phenomena to a so-called “pseudo-piezoelectricity” as summarized in Petrenko & Whitworth (1999). This phenomenon assumed that the polarization in ice is proportional to the pressure gradient. In fact, Petrenko and co-workers studied point defects, dislocations and cracks in ice and arrived at their conclusions about pressure graident dependent polarization from a micro- scopic view point. This, as will be shown later, is a natural result of what is known today as flexoelectricity.
This chapter is organized as follows. First, we construct the Green’s function for a flexoelectric boundary value problem. We will use it in our studies of point defects and dislocations. Second, we give an analytic solution to the problem of a single point defect in an isotropic flexoelectric solid. Third, we solve for the polarization fields of screw and edge dislocations and connect our analysis to various experiments. In particular, we will show in this chapter that some of the results for charged dislocations can be qualitatively understood in terms of flexoelectricity.