Zhigang Suo, Xuanhe Zhao, and William H. Greene

Abstract, Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell-Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both conservative and dissipative dielectrics. As applications of the theory, we discuss material laws for conservative dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.

Introduction

All materials contain electrons and protons. In a dielectric, these charged particles form bonds, and move relative to one another by short distances in response to a voltage or a force. That is, all dielectrics are deformable. The notion of a rigid dielectric is as fictitious as that of a rigid body: they are idealizations useful for some purposes, but misleading for others.

Deformable dielectrics are central in diverse technologies (Newhham, 2005; Uchino, 1997; Sessler, 1987; Campbell, 1998). Our own interest is renewed by recent innovations in materials, principally organics capable of large deformation, including electrostrictive polymers (Zhang, 1998; Chu et al., 2006), cellular electrets (Graz, et al., 2006), liquid crystal elastomers (Warner and Terentjev, 2003), and elastomers capable of large deformation under electric field (e.g., Pelrine, et al, 2000). Also emerging are technologies to produce patterns of electric charge with small features (Jacobs and Whitesides, 2001; McCarty et al., 2006). Potential applications of these materials and technologies include transducers in large-area, flexible electronics (e.g., displays, artificial muscles, and sensitive skins), as well as in devices at small length scales. Phenomena of electric-field induced motion and instability have also been actively studied (e.g., Li and Aluru, 2002; Gao and Suo, 2003; Suo and Hong, 2004; Huang, 2005; Lu and Salac, 2006; Zhu et al., 2006).

Although the atomic origin of dielectric deformation has long been understood, how to formulate a field theory remains controversial. Many theories have been formulated (e.g., Becker, 1982; Landau and Lifshitz, 1984; Toupin, 1956; Eringen, 1963; Pao, 1978; Eringen and Maugin, 1989; Maugin, et al., 1992; Kuang, 2002), invoking different approximations and postulates. On these theories, Pao (1978) remarked, "That there are so many coexisting theories and results for a subject so fundamental in nature may sound very surprising to experimentalists, for theories can usually be sorted out, or proven to be fallacious by carefully designed experiments. The difficulty here is that the electromagnetic fields inside matter are expressed in terms of field variables which cannot be directly measured in laboratories." Recent critiques of these theories may be found in Rinaldi and Brenner (2002), and in McMeeking and Landis (2005).

Pao's remarks were directed to general theories of electromagnetism in matter, but we find that his remarks apt for theories of deformable dielectrics. To give some ideas of the controversies involved, we mention two difficulties.

One difficulty has to do with the notion of electric force. Consider, for example, a parallel-plate capacitor, consisting of an insulating medium and two electrodes, with a battery maintaining a positive charge on one electrode, and a negative charge on the other (Fig. 1). If the insulating medium is a vacuum or a fluid, we must apply a force (e.g., by using a weight) to maintain equilibrium. In this case, there is no ambiguity as to what the electric force is: the force between the two electrodes can be measured by the weight. Maxwell (1891) converted this force into a state of stress in the medium. When the insulating medium is a solid dielectric, however, the electric force cannot be measured. Indeed, for many common solid dielectrics subject to a voltage, the two electrodes appear to repel, rather than attract, each other (Newnham, 2005). The atomic origin of this phenomenon is clear. Influenced by the voltage between the electrodes, charges inside the dielectric tend to displace relative to one another, often accompanied by an elongation of the material in the direction of the electric field.

On the force between electric charges in a solid, Feynman (1964) remarked, "This is a very difficult problem which has not been solved, because it is, in a sense, indeterminate. If you put charges inside a dielectric solid, there are many kinds of pressures and strains. You cannot deal with virtual work without including also the mechanical energy required to compress the solid, and it is a difficult matter, generally speaking, to make a unique distinction between the electrical forces and mechanical forces due to solid material itself. Fortunately, no one ever really needs to know the answer to the question proposed. He may sometimes want to know how much strain there is going to be in a solid, and that can be worked out. But it is much more complicated than the simple result we got for liquids."

A second difficulty has to do with work conjugates. It is a simple matter to show that, when a dielectric solid deforms, the true electric field and the true electric displacement are not work conjugates. Although this fact does not preclude them from being used to formulate the field theory of deformable dielectrics, the non-conjugates do lead to complications, and their almost exclusive, and sometimes erroneous, use in the literature contributes to the controversies.

While the first difficulty was noted by nearly all authors on the subject, we have not found any explicit discussion on the second. The dubious status of electric force is unsettling as most textbooks start by defining electric field by the force acting on a test charge divided by the amount of charge. In a solid dielectric, the electric force is not a measurable quantity, so that this definition is not operational. A common approach is to forgo this definition, and regard the field equations of electrostatics as the starting point. But to discuss deformation one has to link the electric field to a force, and this connection is usually made by invoking work. In this connection, some authors (e.g., Landau and Lifshitz, 1984; Line and Glass, 1977) assumed that the true electric field and the true electric displacement are work conjugates. While this assumption does not lead to serious errors for infinitesimal deformation, it does for finite deformation.

In this paper, in the spirit of Feynman's remark, instead of dwelling on the meaningless notion of electric force, we ask questions concerning measurable quantities. In effect, we ask, given an applied voltage and applied force, how much does one electrode move relative to the other, and how much charge flows from one electrode to the other? Instead of leaving various fields undefined, we define them by operational procedures.

As a mental aid in formulating the theory, imagine that each material particle in a dielectric is attached with a weight and a battery, and then prescribe a field of virtual displacement and a field of virtual voltage. We will use virtual work to define fields inside media, an approach well established in mechanics, but perhaps less so in electrostatics. Associated with the work done by the weights and inertia, we define the stress inside the dielectric as the conjugate to the gradient of displacement. Associated with the work done by the batteries, we define the electric displacement inside the dielectric as the conjugate to the gradient of electric potential. The approach requires no additional postulate beyond what we mean by work, displacement, charge and inertia. The approach does not start with field equations, but produces them as consequences. The theory is applicable to finite deformation, and to both conservative and dissipative dielectrics.

We write the body of the paper with minimal digression, hoping that a reader with basic knowledge of electrostatics and mechanics can appreciate the theory. Section 2 reviews elementary facts of work, energy and electromechanical coupling, using a generic transducer. Section 3 uses a homogenous field to illustrate a procedure to define quantities per unit length, area, and volume, a procedure that we generalize in Section 4 to inhomogeneous fields in three dimensions. Section 5 sketches the material laws for conservative media. Section 6 applies the theory to fluid dielectrics, and recovers the Maxwell stress. Section 7 discusses solid dielectrics. Section 8 applies the theory to infinitesimal fields superimposed upon a given field.

Our approach allows us to choose, among many alternatives, measures of stress, strain, electric field, and electric displacement. The body of the paper will focus on nominal quantities using material coordinates. Various Appendices describe alternative formulations and link to the existing literature. We show that our theory recovers the results of McMeeking and Landis (2005), who formulated a theory of deformable dielectrics by using spatial coordinates and the true electric field and true electric displacement. These authors started with an electric force, but concluded that this force cannot be measured in solid dielectrics. A parallel reading of that paper and the present one should provide a fuller understanding of both approaches.

Update on 4 June 2007.  I attach the slides to be presented on McMat 2007 in Austin, on 6 June 2007.